Road to 800

Geometry

Chapter 13 · 17 lessons · 468 questions

Arcs, sectors and radians

Key points

  • An arc is part of the circumference. A sector is a slice of the circle's area.
  • The fraction of the circle is the central angle over 360∘360^\circ: partwhole\dfrac{\text{part}}{\text{whole}} =angle360= \dfrac{\text{angle}}{360}.
  • Circumference =2πr= 2\pi r. Area =πr2= \pi r^2.
  • 180∘180^\circ is π\pi radians. With the angle θ\theta in radians, the arc length is rθr\theta.

See it solved

Example

A circle has a radius of 99. A central angle of 80∘80^\circ cuts off an arc of the circle. The length of the arc is kπk\pi. What is the value of kk?

Worked by hand

The plan

  1. Set up a proportion
  2. Determine the circumference
  3. Determine the arc

Desmos does not help you set up the proportion. It can only keep the calculation on one row, as a check.

  1. Step 1: Set up a proportion

    • arccircumference\dfrac{\text{arc}}{\text{circumference}} =80360= \dfrac{80}{360}.
    • 80360=29\dfrac{80}{360} = \dfrac{2}{9}. The arc is 29\dfrac{2}{9} of the circumference.

  2. Step 2: Determine the circumference

    • 2πr2\pi r =2⋅π⋅9= 2 \cdot \pi \cdot 9 =18π= 18\pi.

  3. Step 3: Determine the arc

    • 29⋅18π\dfrac{2}{9} \cdot 18\pi =36π9= \dfrac{36\pi}{9} =4π= 4\pi.
    • So k=4k = 4.
The correct answer is 44.

The same example in Desmos

  1. new graph

    Keys: 80/360 → *2*9

    44, at the right of the row. So the arc is 4π4\pi.

Type
Desmos expression list: frac{80}{360}·2·9

Made with Desmos

Open Desmos

Your turn

A circle has a radius of 66. A sector of the circle has a central angle of 120∘120^\circ. The area of the sector is kπk\pi. What is the value of kk?

Show the answerHide the answer

Answer

The correct answer is 1212.

  1. Step 1: Set up a proportion

    • sectorarea of the circle\dfrac{\text{sector}}{\text{area of the circle}} =120360= \dfrac{120}{360} =13= \dfrac{1}{3}.

  2. Step 2: Determine the area of the circle

    • πr2\pi r^2 =π⋅62= \pi \cdot 6^2 =36π= 36\pi.

  3. Step 3: Determine the sector

    • 13⋅36π=12π\dfrac{1}{3} \cdot 36\pi = 12\pi. So k=12k = 12.

Traps

  • Using the area for an arc. 29⋅π⋅92\dfrac{2}{9} \cdot \pi \cdot 9^2 =18π= 18\pi is the area of the sector, not the length of the arc. An arc is part of the circumference.
  • Dividing by 180180 instead of 360360. A full circle is 360∘360^\circ. 80180⋅18π=8π\dfrac{80}{180} \cdot 18\pi = 8\pi is twice the right answer.
Practice this lesson36 questions

Inscribed angles, chords and tangents

Key points

  • A central angle has its vertex at the center of the circle. An inscribed angle has its vertex on the circle.
  • An inscribed angle is half the central angle that cuts off the same arc.
  • A tangent line touches the circle at one point and is perpendicular to the radius at that point.
  • Two radii and a chord make an isosceles triangle, because all radii are equal.

See it solved

Example

In the figure, OO is the center of the circle, and angle AOBAOB measures 110∘110^\circ. What is the measure, in degrees, of angle ACBACB?

OABC110°

Worked by hand

The plan

  1. Identify the types of angles
  2. Halve the central angle

Desmos does not help you name the angles. It can only check the arithmetic.

  1. Step 1: Identify the types of angles

    • Angle AOBAOB has its vertex at the center OO. It is a central angle.
    • Angle ACBACB has its vertex on the circle. It is an inscribed angle.
    • Both angles cut off the same arc, ABAB.

  2. Step 2: Halve the central angle

    • Angle ACB=1102=55ACB = \dfrac{110}{2} = 55 degrees.
The correct answer is 5555.

The same example in Desmos

  1. new graph

    5555, at the right of the row.

Type
Desmos expression list: frac{110}{2}

Made with Desmos

Open Desmos

Your turn

A line is tangent to a circle with center OO at the point PP. The point QQ is on the tangent line, with PQ=12PQ = 12 and OQ=13OQ = 13. What is the radius of the circle?

Show the answerHide the answer

Answer

The correct answer is 55.

  1. Step 1: Identify the right angle

    • A tangent is perpendicular to the radius at the point where it touches.
    • So angle OPQOPQ is a right angle, and OQOQ is the hypotenuse of triangle OPQOPQ.

  2. Step 2: Use the Pythagorean theorem

    • OP2+122OP^2 + 12^2 =132= 13^2.
    • 122=14412^2 = 144 and 132=16913^2 = 169. So OP2OP^2 =169−144= 169 - 144 =25= 25.
    • OP=25=5OP = \sqrt{25} = 5. The radius is 55.

Traps

  • Doubling instead of halving. The inscribed angle is the smaller one. 2⋅110=2202 \cdot 110 = 220 degrees cannot be an angle of a triangle.
  • Treating the radius as the hypotenuse. In Your turn, the right angle is at PP, so the hypotenuse is OQ=13OQ = 13. Adding the squares, 122+13212^2 + 13^2, is the wrong set-up.
Practice this lesson36 questions

The equation of a circle

Key points

  • (x−h)2+(y−k)2(x - h)^2 + (y - k)^2 =r2= r^2 is a circle with center (h,k)(h, k) and radius rr.
  • The signs flip: (x+3)2(x + 3)^2 means h=−3h = -3.
  • The right side is r2r^2, not rr. Take the square root to get the radius.
  • If the equation is written out, complete the square in xx and in yy to get this form.

See it solved

Example

In the xyxy-plane, the graph of x2+8x+y2−2yx^2 + 8x + y^2 - 2y =8= 8 is a circle. What is the radius of the circle?

Worked by hand

The plan

  1. Group the terms
  2. Complete the square in xx and in yy
  3. Read the center and the radius

Desmos is quicker when the equation is written out: type it as given, then click the top and the bottom of the circle.

  1. Step 1: Group the terms

    • The xx terms are together: x2+8xx^2 + 8x.
    • The yy terms are together: y2−2yy^2 - 2y.
    • The constant, 88, is already alone on the right.

  2. Step 2: Complete the square in xx and in yy

    • Half of 88 is 44, and 42=164^2 = 16. Adding 1616 turns x2+8xx^2 + 8x into (x+4)2(x + 4)^2.
    • Half of −2-2 is −1-1, and (−1)2=1(-1)^2 = 1. Adding 11 turns y2−2yy^2 - 2y into (y−1)2(y - 1)^2.
    • Add the same amounts to the right side: 8+16+1=258 + 16 + 1 = 25.
    • (x+4)2+(y−1)2(x + 4)^2 + (y - 1)^2 =25= 25.

  3. Step 3: Read the center and the radius

    • The center is (−4,1)(-4, 1).
    • r2=25r^2 = 25, so r=25=5r = \sqrt{25} = 5.
The correct answer is 55.

The same example in Desmos

  1. new graph

    Keys: x^2 → +8x+y^2 → -2y=8

    The circle. If it looks like an oval, press the home button so the axes have the same scale.

  2. click the circle, then its top and bottom gray points

    (−4,6)(-4, 6) and (−4,−4)(-4, -4). The center is halfway between them, at (−4,1)(-4, 1). The radius is half of 6−(−4)=106 - (-4) = 10, which is 55.

Type
Desmos expression list: x^2+8x+y^2-2y=8
Look
Desmos graph of x^2+8x+y^2-2y=8, with (−4, 6) and (−4, −4) labelled

Made with Desmos

Open Desmos

Your turn

In the xyxy-plane, a circle has the equation (x−2)2+(y+5)2(x - 2)^2 + (y + 5)^2 =49= 49. What are the center and the radius of the circle?

Show the answerHide the answer

Answer

The center is (2,−5)(2, -5) and the radius is 77.

  1. Step 1: Read the center

    • (x−2)2(x - 2)^2 gives h=2h = 2.
    • (y+5)2(y + 5)^2 is (y−(−5))2(y - (-5))^2, so k=−5k = -5.
    • The center is (2,−5)(2, -5).

  2. Step 2: Read the radius

    • r2=49r^2 = 49, so r=49=7r = \sqrt{49} = 7.

Traps

  • Reading r2r^2 as the radius. The right side is 2525, but the radius is 25=5\sqrt{25} = 5. 2525 is a wrong answer.
  • Forgetting to add to both sides. Completing the square adds 1616 and 11 on the left, so add them on the right too. Leaving the right side as 88 gives r2=8r^2 = 8, a wrong answer.
Practice this lesson32 questions

Circles meeting lines and axes

Key points

  • The distance from the center (h,k)(h, k) to the xx-axis is ∣k∣|k|. Its distance to the yy-axis is ∣h∣|h|.
  • A circle is tangent to an axis when that distance equals the radius.
  • A horizontal line is tangent to the circle at its top, y=k+ry = k + r, or at its bottom, y=k−ry = k - r.
  • A point is inside the circle if (x−h)2+(y−k)2(x - h)^2 + (y - k)^2 is less than r2r^2, on the circle if it equals r2r^2, and outside if it is greater.

See it solved

Example

In the xyxy-plane, a circle has its center at (3,−4)(3, -4) and is tangent to the xx-axis. Which equation represents the circle?

  1. A(x−3)2+(y+4)2(x - 3)^2 + (y + 4)^2 =4= 4
  2. B(x−3)2+(y+4)2(x - 3)^2 + (y + 4)^2 =9= 9
  3. C(x−3)2+(y+4)2(x - 3)^2 + (y + 4)^2 =16= 16
  4. D(x+3)2+(y−4)2(x + 3)^2 + (y - 4)^2 =16= 16

Worked by hand

The plan

  1. Determine the radius
  2. Write the equation

With answer choices, Desmos is quicker: type each choice and keep the circle that just touches the axis.

  1. Step 1: Determine the radius

    • The center is (3,−4)(3, -4). Its distance from the xx-axis is ∣−4∣=4|-4| = 4.
    • The circle is tangent to the xx-axis, so the radius is 44.

  2. Step 2: Write the equation

    • r2=42=16r^2 = 4^2 = 16. That rules out choices A and B.
    • The center (3,−4)(3, -4) gives (x−3)2(x - 3)^2 and (y+4)2(y + 4)^2. That rules out choice D.
    • (x−3)2+(y+4)2(x - 3)^2 + (y + 4)^2 =16= 16.
The correct answer choice is C: (x−3)2+(y+4)2(x - 3)^2 + (y + 4)^2 =16= 16.

The same example in Desmos

  1. new graph

    Keys: (x-3)^2 → +(y+4)^2 → =16

    The circle for choice C. It just touches the xx-axis.

  2. click the circle, then its top gray point

    (3,0)(3, 0): the one point where the circle meets the xx-axis.

Type
Desmos expression list: (x-3)^2+(y+4)^2=16
Look
Desmos graph of (x-3)^2+(y+4)^2=16, with (3, 0) labelled

Made with Desmos

Open Desmos

Your turn

In the xyxy-plane, the line y=cy = c is tangent to the circle (x−1)2+(y−2)2(x - 1)^2 + (y - 2)^2 =25= 25, where cc is a positive constant. What is the value of cc?

Show the answerHide the answer

Answer

The correct answer is 77.

  1. Step 1: Read the center and the radius

    • The center is (1,2)(1, 2).
    • r2=25r^2 = 25, so r=5r = 5.

  2. Step 2: Determine the top and the bottom of the circle

    • Top: y=2+5=7y = 2 + 5 = 7.
    • Bottom: y=2−5=−3y = 2 - 5 = -3.
    • A horizontal tangent line is at the top or the bottom. cc is positive, so c=7c = 7.

Traps

  • Using the xx-coordinate for the distance to the xx-axis. That distance is how far the center is above or below the axis: ∣−4∣=4|-4| = 4. Using 33 gives r2=9r^2 = 9, choice B, a wrong answer.
  • Writing rr instead of r2r^2. The right side is 42=164^2 = 16, not 44. Choice A is a wrong answer.
Practice this lesson36 questions

Angles and parallel lines

Key points

  • Angles on a straight line add to 180∘180^\circ. Vertical angles, opposite each other where two lines cross, are equal.
  • When a line crosses two parallel lines, corresponding angles are equal and alternate interior angles are equal.
  • Same-side interior angles add to 180∘180^\circ.
  • So every angle at the two crossings is one of two sizes, and the two sizes add to 180∘180^\circ.

See it solved

Example

In the figure, lines ℓ\ell and mm are parallel. What is the value of xx?

(2x + 10)°(3x − 5)°ℓm

Note: Figure not drawn to scale.

Worked by hand

The plan

  1. Identify the types of angles
  2. Set up and solve the equation

Desmos does not help you name the angle pair. Once the equation is written, typing it is only a check.

  1. Step 1: Identify the types of angles

    • The two marked angles are in the same position at each crossing: above the parallel line and to the right of the third line.
    • They are corresponding angles, so they are equal.

  2. Step 2: Set up and solve the equation

    • 2x+10=3x−52x + 10 = 3x - 5.
    • Subtract 2x2x from both sides: 10=x−510 = x - 5.
    • Add 55 to both sides: x=15x = 15.
    • Check: 2⋅15+10=402 \cdot 15 + 10 = 40 and 3⋅15−5=403 \cdot 15 - 5 = 40. Both angles are 40∘40^\circ.
The correct answer is 1515.

The same example in Desmos

  1. new graph

    A vertical line.

  2. zoom out, then click where the line meets the x-axis

    (15,0)(15, 0). So x=15x = 15.

Type
Desmos expression list: 2x+10=3x-5
Look
Desmos graph of 2x+10=3x-5, with (15, 0) labelled

Made with Desmos

Open Desmos

Your turn

Lines pp and qq are parallel, and a third line crosses both of them. Two same-side interior angles measure (4x+20)∘(4x + 20)^\circ and (2x+10)∘(2x + 10)^\circ. What is the value of xx?

Show the answerHide the answer

Answer

The correct answer is 2525.

  1. Step 1: Identify the types of angles

    • Same-side interior angles add to 180∘180^\circ. They are not equal.

  2. Step 2: Set up and solve the equation

    • Add the two angles and set the sum equal to 180180.
    • 4x+2x=6x4x + 2x = 6x and 20+10=3020 + 10 = 30.
    • So 6x+30=1806x + 30 = 180.
    • Subtract 3030 from both sides: 6x=1506x = 150.
    • x=1506=25x = \dfrac{150}{6} = 25.

Traps

  • Setting same-side interior angles equal. They add to 180∘180^\circ. 4x+204x + 20 =2x+10= 2x + 10 gives x=−5x = -5, which would make the first angle 0∘0^\circ.
  • Adding corresponding angles to 180180. Corresponding angles are equal. Adding them gives 5x+5=1805x + 5 = 180, so 5x=1755x = 175 and x=35x = 35, a wrong answer.
Practice this lesson32 questions

Polygon angle sums

Key points

  • The interior angles of a polygon with nn sides add to 180(n−2)180(n - 2) degrees.
  • A triangle has 180∘180^\circ, a quadrilateral 360∘360^\circ and a pentagon 540∘540^\circ.
  • The exterior angles of any polygon add to 360∘360^\circ.
  • In a regular polygon all the angles are equal, so divide the sum by nn.

See it solved

Example

What is the measure, in degrees, of each interior angle of a regular octagon?

Worked by hand

The plan

  1. Determine the angle sum
  2. Determine one angle

The formula takes one line by hand. Desmos only keeps the calculation on one row, as a check.

  1. Step 1: Determine the angle sum

    • An octagon has n=8n = 8 sides.
    • 180(8−2)180(8 - 2) =180⋅6= 180 \cdot 6 =1,080= 1{,}080 degrees.

  2. Step 2: Determine one angle

    • A regular octagon has 88 equal angles.
    • 1,0808=135\dfrac{1{,}080}{8} = 135 degrees.
The correct answer is 135135.

The same example in Desmos

  1. new graph

    Keys: 180(8-2)/8

    135135, at the right of the row.

Type
Desmos expression list: frac{180(8-2)}{8}

Made with Desmos

Open Desmos

Your turn

Four of the interior angles of a pentagon measure 100∘100^\circ, 110∘110^\circ, 95∘95^\circ and 120∘120^\circ. What is the measure, in degrees, of the fifth angle?

Show the answerHide the answer

Answer

The correct answer is 115115.

  1. Step 1: Determine the angle sum

    • A pentagon has n=5n = 5 sides.
    • 180(5−2)180(5 - 2) =180⋅3= 180 \cdot 3 =540= 540 degrees.

  2. Step 2: Determine the missing angle

    • Add the four known angles: 100+110=210100 + 110 = 210.
    • 210+95=305210 + 95 = 305.
    • 305+120=425305 + 120 = 425.
    • The fifth angle is 540−425=115540 - 425 = 115 degrees.

Traps

  • Using 180n180n. 180⋅8=1,440180 \cdot 8 = 1{,}440 forgets the −2-2. It would make each angle 180∘180^\circ, which is a straight line, not a corner.
  • Assuming every pentagon is regular. 5405=108\dfrac{540}{5} = 108 only works when all five angles are equal. Here they are not: subtract the four known angles from 540540.
Practice this lesson12 questions

Triangle angles and sides

Key points

  • The three angles of a triangle add to 180∘180^\circ.
  • An exterior angle equals the sum of the two interior angles that are not next to it.
  • An isosceles triangle has two equal sides, and the angles opposite those sides are equal.
  • Any side of a triangle is shorter than the sum of the other two sides and longer than their difference.

See it solved

Example

In the figure, AB=ACAB = AC and angle AA measures 40∘40^\circ. What is the measure, in degrees, of angle BB?

ABC40°

Worked by hand

The plan

  1. Identify the equal angles
  2. Use the angle sum

Desmos does not help you see which angles are equal. It can only check the arithmetic.

  1. Step 1: Identify the equal angles

    • AB=ACAB = AC, so the triangle is isosceles.
    • The angles opposite the equal sides are equal: angle BB equals angle CC.

  2. Step 2: Use the angle sum

    • The three angles add to 180∘180^\circ, and angle AA is 40∘40^\circ.
    • Angles BB and CC together are 180−40=140180 - 40 = 140 degrees.
    • They are equal, so angle BB is 1402=70\dfrac{140}{2} = 70 degrees.
The correct answer is 7070.

The same example in Desmos

  1. new graph

    7070, at the right of the row.

Type
Desmos expression list: frac{180-40}{2}

Made with Desmos

Open Desmos

Your turn

Two sides of a triangle have lengths 55 and 99. The length of the third side is a whole number. What is the greatest possible length of the third side?

Show the answerHide the answer

Answer

The correct answer is 1313.

  1. Step 1: Determine the limits for the third side

    • The third side must be less than the sum of the other two: 5+9=145 + 9 = 14.
    • It must be more than their difference: 9−5=49 - 5 = 4.
    • So the third side is between 44 and 1414.

  2. Step 2: Determine the greatest whole number

    • 1414 itself is not allowed. The greatest whole number less than 1414 is 1313.

Traps

  • Subtracting only once. 180−40=140180 - 40 = 140 is the sum of the two base angles, not one of them. Halve it: 70∘70^\circ.
  • Including the boundary. A third side of exactly 1414 would make the triangle flat, a straight line. The side must be less than 1414, so the greatest whole number is 1313.
Practice this lesson20 questions

Pythagorean theorem and special right triangles

Key points

  • In a right triangle, a2+b2=c2a^2 + b^2 = c^2. Here cc is the hypotenuse: the longest side, opposite the right angle.
  • Common triples are 33-44-55, 55-1212-1313 and 88-1515-1717, and their multiples, such as 66-88-1010.
  • In a 45∘45^\circ-45∘45^\circ-90∘90^\circ triangle, the sides are xx, xx and x2x\sqrt{2}.
  • In a 30∘30^\circ-60∘60^\circ-90∘90^\circ triangle, the sides are xx, x3x\sqrt{3} and 2x2x. The shortest side is opposite the 30∘30^\circ angle.

See it solved

Example

A right triangle has legs of length 99 and 1212. What is the length of its hypotenuse?

Worked by hand

The plan

  1. Look for a triple
  2. Check with the Pythagorean theorem

Spotting a triple is quicker than any calculator. Desmos can only check the square root.

  1. Step 1: Look for a triple

    • 9=3⋅39 = 3 \cdot 3 and 12=3⋅412 = 3 \cdot 4.
    • This is the 33-44-55 triple with every side multiplied by 33.
    • The hypotenuse is 3⋅5=153 \cdot 5 = 15.

  2. Step 2: Check with the Pythagorean theorem

    • 92+1229^2 + 12^2 =81+144= 81 + 144 =225= 225.
    • 225=15\sqrt{225} = 15. It matches.
The correct answer is 1515.

The same example in Desmos

  1. new graph

    Keys: sqrt 9^2 → +12^2

    1515, at the right of the row.

Type
Desmos expression list: sqrt{9^2+12^2}

Made with Desmos

Open Desmos

Your turn

In a right triangle with angles of 30∘30^\circ, 60∘60^\circ and 90∘90^\circ, the hypotenuse has length 1010. What is the length of the side opposite the 60∘60^\circ angle?

Show the answerHide the answer

Answer

The correct answer is 535\sqrt{3}.

  1. Step 1: Match the pattern

    • The sides are xx, x3x\sqrt{3} and 2x2x. The hypotenuse is 2x2x.
    • 2x=102x = 10, so x=5x = 5.

  2. Step 2: Determine the side asked for

    • The side opposite the 60∘60^\circ angle is the middle one, x3x\sqrt{3}.
    • With x=5x = 5, that is 535\sqrt{3}.

Traps

  • Adding the legs. 9+12=219 + 12 = 21 is not the hypotenuse. The squares add, not the sides: 81+144=22581 + 144 = 225, and 225=15\sqrt{225} = 15.
  • Putting 3\sqrt{3} on the hypotenuse. In a 30∘30^\circ-60∘60^\circ-90∘90^\circ triangle the hypotenuse is 2x2x, with no root. 10310\sqrt{3} is a wrong answer.
Practice this lesson28 questions

The altitude to the hypotenuse

Key points

  • The altitude from the right angle to the hypotenuse splits a right triangle into two smaller right triangles.
  • All three triangles are similar: the big one and the two small ones have the same angles.
  • The altitude squared equals the product of the two pieces of the hypotenuse: h2=p⋅qh^2 = p \cdot q.
  • Match corresponding sides carefully: short leg with short leg, long leg with long leg.

See it solved

Example

In the figure, angle ACBACB is a right angle and CDCD is perpendicular to ABAB. What is the length of CDCD?

ABCD49

Worked by hand

The plan

  1. Identify the similar triangles
  2. Set up a proportion for the corresponding sides

Desmos does not help you match the sides. It can only check the square root.

  1. Step 1: Identify the similar triangles

    • Triangles ADCADC and CDBCDB each have a right angle at DD.
    • Angle AA and angle DCBDCB are equal, because each one is 90∘90^\circ minus angle BB.
    • Two pairs of equal angles: triangle ADCADC is similar to triangle CDBCDB.

  2. Step 2: Set up a proportion for the corresponding sides

    • In triangle ADCADC, the short leg is AD=4AD = 4 and the long leg is CDCD.
    • In triangle CDBCDB, the short leg is CDCD and the long leg is DB=9DB = 9.
    • Short leg over long leg: 4CD=CD9\dfrac{4}{CD} = \dfrac{CD}{9}.
    • Cross-multiply: CD2=4⋅9CD^2 = 4 \cdot 9, so CD2=36CD^2 = 36.
    • CD=36=6CD = \sqrt{36} = 6.
The correct answer is 66.

The same example in Desmos

  1. new graph

    66, at the right of the row: the altitude.

Type
Desmos expression list: sqrt{4·9}

Made with Desmos

Open Desmos

Your turn

In a right triangle, the altitude to the hypotenuse divides the hypotenuse into two pieces of length 99 and 1616. What is the length of the altitude?

Show the answerHide the answer

Answer

The correct answer is 1212.

  1. Step 1: Use the altitude rule

    • The altitude squared equals the product of the two pieces: h2=9⋅16h^2 = 9 \cdot 16.
    • 9⋅16=1449 \cdot 16 = 144, so h2=144h^2 = 144.
    • h=144=12h = \sqrt{144} = 12.

Traps

  • Averaging the two pieces. 4+92=6.5\dfrac{4 + 9}{2} = 6.5 is not the altitude. Multiply the pieces and take the square root: 36=6\sqrt{36} = 6.
  • Forgetting the square root. 4⋅9=364 \cdot 9 = 36 is CD2CD^2, not CDCD. The altitude is 66.
Practice this lesson24 questions

Similar and congruent triangles

Key points

  • Similar triangles have the same angles, and their sides are in proportion.
  • Two triangles are similar if two angles of one equal two angles of the other.
  • A line parallel to one side of a triangle cuts off a smaller triangle that is similar to the whole triangle.
  • Match corresponding sides: they lie opposite equal angles. Then set up a proportion.

See it solved

Example

In the figure, DEDE is parallel to BCBC. What is the length of BCBC?

ABCDE863

Worked by hand

The plan

  1. Identify the similar triangles
  2. Set up a proportion for the corresponding sides

Desmos does not help you match the sides. Once the proportion is written, typing it is only a check.

  1. Step 1: Identify the similar triangles

    • DEDE is parallel to BCBC, so angle ADEADE equals angle ABCABC.
    • Triangles ADEADE and ABCABC also share angle AA.
    • Two pairs of equal angles: triangle ADEADE is similar to triangle ABCABC.

  2. Step 2: Set up a proportion for the corresponding sides

    • ADAD matches ABAB, and DEDE matches BCBC.
    • ABAB is the whole side: AB=6+3=9AB = 6 + 3 = 9.
    • Small over big on both sides: 69=8BC\dfrac{6}{9} = \dfrac{8}{BC}.
    • Cross-multiply: 6⋅BC=9⋅86 \cdot BC = 9 \cdot 8, so 6⋅BC=726 \cdot BC = 72.
    • BC=726=12BC = \dfrac{72}{6} = 12.
The correct answer is 1212.

The same example in Desmos

  1. new graph

    Keys: 6/9 → =8/x

    A vertical line, with xx for the length of BCBC.

  2. zoom out, then read where the line meets the x-axis

    x=12x = 12. So BC=12BC = 12.

Type
Desmos expression list: frac{6}{9}=frac{8}{x}
Look
Desmos graph of frac{6}{9}=frac{8}{x}, with (12, 0) labelled

Made with Desmos

Open Desmos

Your turn

A pole that is 66 feet tall casts a shadow that is 44 feet long. At the same time, a tree casts a shadow that is 1414 feet long. How tall is the tree, in feet?

Show the answerHide the answer

Answer

The correct answer is 2121.

  1. Step 1: Identify the similar triangles

    • The sun's rays meet the ground at the same angle for the pole and for the tree.
    • Each one makes a right triangle with its shadow, so the two triangles are similar.

  2. Step 2: Set up a proportion for the corresponding sides

    • Height over shadow on both sides: 64=h14\dfrac{6}{4} = \dfrac{h}{14}.
    • Cross-multiply: 4h=6⋅14=844h = 6 \cdot 14 = 84.
    • h=844=21h = \dfrac{84}{4} = 21 feet.

Traps

  • Using part of a side. 63=8BC\dfrac{6}{3} = \dfrac{8}{BC} matches ADAD with DBDB, but DBDB is not a side of triangle ABCABC. It gives BC=4BC = 4, which is shorter than DEDE, a wrong answer.
  • Mixing the order. With the small triangle on top on the left, it must be on top on the right too. 69=BC8\dfrac{6}{9} = \dfrac{BC}{8} gives BC=163BC = \dfrac{16}{3}, a wrong answer.
Practice this lesson52 questions

Squares, rectangles, cubes and boxes

Key points

  • Rectangle: area = length ×\times width, and perimeter = 2×2 \times length + 2×+\ 2 \times width.
  • Box: volume = length ×\times width ×\times height.
  • A cube with side ss has volume s3s^3 and surface area 6s26s^2, because it has six equal square faces.
  • To go back from an area or a volume to a side, take the square root or the cube root. A length is positive.

See it solved

Example

A rectangle has an area of 8484 square inches and a width of 77 inches. What is the perimeter of the rectangle, in inches?

Worked by hand

The plan

  1. Determine the length
  2. Determine the perimeter

Desmos does not help you choose the formula. Once the equation is written, typing it is only a check.

  1. Step 1: Determine the length

    • Area = length ×\times width, so 84=l⋅784 = l \cdot 7.
    • l=847=12l = \dfrac{84}{7} = 12 inches.

  2. Step 2: Determine the perimeter

    • Perimeter =2⋅12+2⋅7= 2 \cdot 12 + 2 \cdot 7.
    • 2⋅12=242 \cdot 12 = 24 and 2⋅7=142 \cdot 7 = 14.
    • 24+14=3824 + 14 = 38 inches.
The correct answer is 3838.

The same example in Desmos

  1. new graph

    A vertical line, with xx for the length.

  2. zoom out, then click where the line meets the x-axis

    (12,0)(12, 0): the length is 1212 inches.

  3. 3838, at the right of the row: the perimeter.

Type
Desmos expression list: 7x=84; 2·12+2·7
Look
Desmos graph of 7x=84; 2·12+2·7, with (12, 0) labelled

Made with Desmos

Open Desmos

Your turn

A cube has a volume of 6464 cubic centimeters. What is the surface area of the cube, in square centimeters?

Show the answerHide the answer

Answer

The correct answer is 9696.

  1. Step 1: Determine the side

    • s3=64s^3 = 64.
    • 4⋅4=164 \cdot 4 = 16, and 16⋅4=6416 \cdot 4 = 64. So s=4s = 4 centimeters.

  2. Step 2: Determine the surface area

    • One face: 42=164^2 = 16 square centimeters.
    • Six faces: 6⋅16=966 \cdot 16 = 96 square centimeters.

Traps

  • Stopping at the length. 1212 inches is the length of the rectangle. The question asks for the perimeter, 3838 inches.
  • Dividing the volume by 33 to get the side. The side is the cube root: the number that gives 6464 when it is multiplied by itself three times. That is 44, not 643\dfrac{64}{3}.
Practice this lesson20 questions

Triangles, trapezoids and parallelograms: find the height

Key points

  • Triangle: area =12⋅= \dfrac{1}{2} \cdot base ⋅\cdot height. Parallelogram: area = base ⋅\cdot height. Trapezoid: area =12(b1+b2)⋅= \dfrac{1}{2}(b_1 + b_2) \cdot height.
  • The height is always perpendicular to the base. A slanted side is not the height.
  • If the height is missing, draw it. It makes a right triangle, so use the Pythagorean theorem or a triple.
  • Triangles with the same height have areas in the ratio of their bases.

See it solved

Example

In the figure, triangle ABCABC is isosceles, with AC=BC=13AC = BC = 13 and AB=10AB = 10. What is the area of triangle ABCABC?

ABC131310

Worked by hand

The plan

  1. Determine the height
  2. Determine the area

Desmos does not help you find the height in a picture. It can only check the arithmetic.

  1. Step 1: Determine the height

    • Draw the height from CC straight down to the base ABAB. Call the point where it lands MM.
    • The triangle is isosceles, so the height cuts the base in half: AM=102=5AM = \dfrac{10}{2} = 5.
    • Triangle AMCAMC is a right triangle with hypotenuse 1313 and one leg 55.
    • 52+h2=1325^2 + h^2 = 13^2, so 25+h2=16925 + h^2 = 169.
    • h2h^2 =169−25= 169 - 25 =144= 144, so h=12h = 12.

  2. Step 2: Determine the area

    • Area =12⋅10⋅12= \dfrac{1}{2} \cdot 10 \cdot 12.
    • 12⋅10=5\dfrac{1}{2} \cdot 10 = 5, and 5⋅12=605 \cdot 12 = 60.
The correct answer is 6060.

The same example in Desmos

  1. new graph

    Keys: sqrt 13^2 → -5^2

    1212: the height.

  2. Keys: 1/2 → *10*12

    6060: the area.

Type
Desmos expression list: sqrt{13^2-5^2}; frac{1}{2}·10·12

Made with Desmos

Open Desmos

Your turn

A trapezoid has parallel sides of length 88 and 1414. The distance between the parallel sides is 55. What is the area of the trapezoid?

Show the answerHide the answer

Answer

The correct answer is 5555.

  1. Step 1: Determine the height

    • The height is the distance between the parallel sides. It is given: 55.

  2. Step 2: Determine the area

    • Area =12(8+14)⋅5= \dfrac{1}{2}(8 + 14) \cdot 5.
    • 8+14=228 + 14 = 22, and 12⋅22=11\dfrac{1}{2} \cdot 22 = 11.
    • 11⋅5=5511 \cdot 5 = 55.

Traps

  • Using a slanted side as the height. 12⋅10⋅13=65\dfrac{1}{2} \cdot 10 \cdot 13 = 65 uses the side of length 1313, which is not perpendicular to the base. 6565 is a wrong answer.
  • Forgetting the 12\dfrac{1}{2}. 10⋅12=12010 \cdot 12 = 120 is the area of a rectangle around the triangle, twice the right answer.
Practice this lesson20 questions

Volumes of cylinders, cones, spheres, pyramids and prisms

Key points

  • The volume formulas are on the reference sheet. Your job is to choose the right one and plug in.
  • Cylinder: V=πr2hV = \pi r^2 h. Cone: V=13πr2hV = \dfrac{1}{3}\pi r^2 h. Sphere: V=43πr3V = \dfrac{4}{3}\pi r^3.
  • Prism: VV = area of the base ×\times height. Pyramid: V=13×V = \dfrac{1}{3} \times area of the base ×\times height.
  • Turn a diameter into a radius first: r=d2r = \dfrac{d}{2}.

See it solved

Example

A cylinder has a diameter of 1010 centimeters and a height of 88 centimeters. The volume of the cylinder is kπk\pi cubic centimeters. What is the value of kk?

Worked by hand

The plan

  1. List the known values
  2. Plug the values into the formula

Plugging in by hand takes two lines. Desmos is a check: it shows the volume as a decimal, which helps when the choices are decimals.

  1. Step 1: List the known values

    • The diameter is 1010, so the radius is r=102=5r = \dfrac{10}{2} = 5.
    • The height is h=8h = 8.

  2. Step 2: Plug the values into the formula

    • VV =πr2h= \pi r^2 h =π⋅52⋅8= \pi \cdot 5^2 \cdot 8.
    • 52=255^2 = 25, and 25⋅8=20025 \cdot 8 = 200.
    • V=200πV = 200\pi cubic centimeters, so k=200k = 200.
The correct answer is 200200.

The same example in Desmos

  1. new graph

    Keys: pi*5^2 → *8

    About 628.32628.32: the volume as a decimal.

  2. The same decimal. So the volume is 200π200\pi, and k=200k = 200.

Type
Desmos expression list: pi·5^2·8; 200pi

Made with Desmos

Open Desmos

Your turn

A cone has a volume of 48π48\pi cubic inches and a radius of 44 inches. What is the height of the cone, in inches?

Show the answerHide the answer

Answer

The correct answer is 99.

  1. Step 1: List the known values

    • V=48πV = 48\pi and r=4r = 4. The height hh is unknown.

  2. Step 2: Plug the values into the formula

    • V=13πr2hV = \dfrac{1}{3}\pi r^2 h, so 48π48\pi =13π⋅42⋅h= \dfrac{1}{3}\pi \cdot 4^2 \cdot h.
    • 42=164^2 = 16, so 48π=163πh48\pi = \dfrac{16}{3}\pi h.
    • Divide both sides by π\pi: 48=163h48 = \dfrac{16}{3}h.
    • Multiply both sides by 33: 144=16h144 = 16h.
    • h=14416=9h = \dfrac{144}{16} = 9.

Traps

  • Using the diameter as the radius. π⋅102⋅8\pi \cdot 10^2 \cdot 8 =800π= 800\pi is four times too big. Halve the diameter first.
  • Forgetting the 13\dfrac{1}{3} for a cone. 48π=π⋅16⋅h48\pi = \pi \cdot 16 \cdot h gives h=3h = 3, a wrong answer.
Practice this lesson16 questions

Density and other derived units

Key points

  • Density is mass divided by volume.
  • Rearranged: mass = density ×\times volume, and volume = mass ÷\div density.
  • The units tell you what to do. ‘Grams per cubic centimeter’ is grams divided by cubic centimeters.
  • If the volume is not given, find it first with a volume formula.
© Road to 800Also on YouTube

See it solved

Example

A metal block is a rectangular box that measures 55 centimeters by 44 centimeters by 22 centimeters. The metal has a density of 77 grams per cubic centimeter. What is the mass of the block, in grams?

Worked by hand

The plan

  1. Determine the volume
  2. Determine the mass

Desmos does not help you choose the formula. It can only keep the multiplication on one row, as a check.

  1. Step 1: Determine the volume

    • The volume of a box is length times width times height.
    • 5⋅4=205 \cdot 4 = 20, and 20⋅2=4020 \cdot 2 = 40 cubic centimeters.

  2. Step 2: Determine the mass

    • Mass = density ×\times volume.
    • 7⋅40=2807 \cdot 40 = 280 grams.
The correct answer is 280280.

The same example in Desmos

  1. new graph

    280280, at the right of the row: the density times the three sides.

Type
Desmos expression list: 7·5·4·2

Made with Desmos

Open Desmos

Your turn

A stone has a mass of 450450 grams. Its density is 2.52.5 grams per cubic centimeter. What is the volume of the stone, in cubic centimeters?

Show the answerHide the answer

Answer

The correct answer is 180180.

  1. Step 1: Rearrange the formula

    • Density = mass ÷\div volume, so volume = mass ÷\div density.

  2. Step 2: Determine the volume

    • 4502.5=180\dfrac{450}{2.5} = 180 cubic centimeters.
    • Check: 2.5⋅180=4502.5 \cdot 180 = 450 grams. It matches.

Traps

  • Multiplying when you should divide. Mass times density, 450⋅2.5450 \cdot 2.5 =1,125= 1{,}125, has no meaning here. Volume is mass divided by density.
  • Using an area instead of the volume. 5⋅4=205 \cdot 4 = 20 is the area of one face. 7⋅20=1407 \cdot 20 = 140 is a wrong answer: multiply by the height too.
Practice this lesson20 questions

Composite figures and shaded regions

Key points

  • Break the figure into shapes you know: rectangles, triangles and circles.
  • Shaded area = area of the whole −- area of the part that is cut out.
  • Write the plan in words before you calculate.
  • Leave π\pi in the answer unless the question asks for a decimal.

See it solved

Example

In the figure, a circle is drawn inside a square with side length 1010. The circle touches all four sides of the square. What is the area of the region that is inside the square but outside the circle?

10

Worked by hand

The plan

  1. Determine the approach
  2. Determine the area of the square
  3. Determine the area of the circle
  4. Combine the pieces

Desmos does not help you see the pieces. It can only turn the final answer into a decimal, as a check.

  1. Step 1: Determine the approach

    • Area of the region = area of the square −- area of the circle.

  2. Step 2: Determine the area of the square

    • 10⋅10=10010 \cdot 10 = 100.

  3. Step 3: Determine the area of the circle

    • The circle touches all four sides, so its diameter is the side of the square, 1010.
    • Its radius is 102=5\dfrac{10}{2} = 5.
    • Area =πr2= \pi r^2 =π⋅52= \pi \cdot 5^2 =25π= 25\pi.

  4. Step 4: Combine the pieces

    • 100−25π100 - 25\pi.
The correct answer is 100−25π100 - 25\pi.

The same example in Desmos

  1. new graph

    About 21.4621.46: the area as a decimal, in case the choices are decimals.

Type
Desmos expression list: 100-25pi

Made with Desmos

Open Desmos

Your turn

A rectangular garden is 1212 meters long and 88 meters wide. A triangular flower bed with a base of 66 meters and a height of 88 meters is cut out of the garden. What area of the garden is left, in square meters?

Show the answerHide the answer

Answer

The correct answer is 7272.

  1. Step 1: Determine the approach

    • Area left = area of the rectangle −- area of the triangle.

  2. Step 2: Determine the area of the rectangle

    • 12⋅8=9612 \cdot 8 = 96.

  3. Step 3: Determine the area of the triangle

    • 12⋅6⋅8\dfrac{1}{2} \cdot 6 \cdot 8. 12⋅6=3\dfrac{1}{2} \cdot 6 = 3, and 3⋅8=243 \cdot 8 = 24.

  4. Step 4: Combine the pieces

    • 96−24=7296 - 24 = 72 square meters.

Traps

  • Using the side as the radius. π⋅102=100π\pi \cdot 10^2 = 100\pi is the area of a circle with radius 1010, which is bigger than the square itself. The radius is half the side: 55.
  • Subtracting the wrong way round. 25π−10025\pi - 100 is negative, and an area cannot be negative. Take the cut-out part away from the whole.
Practice this lesson16 questions

Figures in the xy-plane

Key points

  • A horizontal or vertical length is a subtraction of coordinates.
  • For a slanted length, make a right triangle. Its legs are the change in xx and the change in yy, and the slanted length is the hypotenuse: d=a2+b2d = \sqrt{a^2 + b^2}.
  • The midpoint of two points averages their coordinates: (x1+x22,y1+y22)\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right).
  • A quick sketch of the points shows which sides are horizontal, vertical or slanted.

See it solved

Example

In the xyxy-plane, a triangle has vertices A(1,2)A(1, 2), B(7,2)B(7, 2) and C(7,10)C(7, 10). What is the perimeter of the triangle?

Worked by hand

The plan

  1. Determine the horizontal and vertical sides
  2. Determine the slanted side
  3. Add the sides

By hand takes a few lines. Desmos is a check, and it draws the sketch for you: plot the points and ask for a distance.

  1. Step 1: Determine the horizontal and vertical sides

    • AA and BB have the same yy-coordinate, so ABAB is horizontal: AB=7−1=6AB = 7 - 1 = 6.
    • BB and CC have the same xx-coordinate, so BCBC is vertical: BC=10−2=8BC = 10 - 2 = 8.

  2. Step 2: Determine the slanted side

    • AC=62+82AC = \sqrt{6^2 + 8^2}.
    • 62=366^2 = 36 and 82=648^2 = 64, and 36+64=10036 + 64 = 100.
    • AC=100=10AC = \sqrt{100} = 10.

  3. Step 3: Add the sides

    • 6+8+10=246 + 8 + 10 = 24.
The correct answer is 2424.

The same example in Desmos

  1. new graph

    The triangle. One side is horizontal and one is vertical.

  2. 1010: the slanted side.

  3. 2424: the perimeter.

Type
Desmos expression list: operatorname{polygon}((1,2),(7,2),(7,10)); operatorname{distance}((1,2),(7,10)); 6+8+10
Look
Desmos graph of operatorname{polygon}((1,2),(7,2),(7,10)); operatorname{distance}((1,2),(7,10)); 6+8+10

Made with Desmos

Open Desmos

Your turn

In the xyxy-plane, a line segment has endpoints (−2,5)(-2, 5) and (6,−1)(6, -1). What is the midpoint of the segment?

Show the answerHide the answer

Answer

The correct answer is (2,2)(2, 2).

  1. Step 1: Average the xx-coordinates

    • −2+62=42=2\dfrac{-2 + 6}{2} = \dfrac{4}{2} = 2.

  2. Step 2: Average the yy-coordinates

    • 5+(−1)2=42=2\dfrac{5 + (-1)}{2} = \dfrac{4}{2} = 2.
    • The midpoint is (2,2)(2, 2).

Traps

  • Adding the legs to get the slanted side. 6+8=146 + 8 = 14 is not the length of ACAC. Square, add, then take the square root: 1010.
  • Subtracting for a midpoint. 6−(−2)2=4\dfrac{6 - (-2)}{2} = 4 is half the distance between the xx-values, not the middle. A midpoint adds the coordinates before halving.
Practice this lesson12 questions

Scaling lengths, areas and volumes

Key points

  • If every length is multiplied by kk, every area is multiplied by k2k^2 and every volume by k3k^3.
  • Doubling the sides makes the area 44 times as large and the volume 88 times as large.
  • Going backwards: from an area ratio take the square root, and from a volume ratio take the cube root, to get the length ratio.
  • For a percent change in a length, turn it into a multiplier first.

See it solved

Example

Triangle PP is similar to triangle QQ. Each side of triangle QQ is 33 times as long as the corresponding side of triangle PP. The area of triangle PP is 1414. What is the area of triangle QQ?

Worked by hand

The plan

  1. Determine the scale factor
  2. Determine the new area

Desmos does not tell you whether to use kk, k2k^2 or k3k^3. It can only check the arithmetic.

  1. Step 1: Determine the scale factor

    • Every length is multiplied by k=3k = 3.

  2. Step 2: Determine the new area

    • Areas are multiplied by k2=32=9k^2 = 3^2 = 9.
    • 14⋅9=12614 \cdot 9 = 126.
The correct answer is 126126.

The same example in Desmos

  1. new graph

    126126, at the right of the row: the area times k2k^2.

Type
Desmos expression list: 14·3^2

Made with Desmos

Open Desmos

Your turn

Each edge of a cube is increased by 50%50\%. The volume of the new cube is how many times the volume of the original cube?

Show the answerHide the answer

Answer

The correct answer is 3.3753.375.

  1. Step 1: Determine the scale factor

    • A 50%50\% increase multiplies each edge by 1+0.50=1.51 + 0.50 = 1.5. So k=1.5k = 1.5.

  2. Step 2: Determine the new volume

    • Volumes are multiplied by k3=1.53k^3 = 1.5^3.
    • 1.5⋅1.5=2.251.5 \cdot 1.5 = 2.25, and 2.25⋅1.52.25 \cdot 1.5 =3.375= 3.375.

Traps

  • Multiplying the area by kk. 14⋅3=4214 \cdot 3 = 42 scales the area as if it were a length. Areas use k2k^2: 14⋅9=12614 \cdot 9 = 126.
  • Reading a 50%50\% increase in the edges as a 50%50\% increase in the volume. The volume is multiplied by 3.3753.375, not by 1.51.5.
Practice this lesson56 questions
Typing tips for Desmos
  • In the keys lines, → means the right-arrow key.
  • Typing ^ or sqrt works the same way: press → to come back down from an exponent or out of a square root before typing the rest.
  • Check the angle mode before any trig: the wrench (Graph Settings) has Radians and Degrees. sin(30) = 0.5 means you are in degrees.
  • Typing / starts a fraction, and everything after it goes into the denominator until you press →. So 42/120*100 typed straight through becomes 42 ÷ (120 × 100). Type 42/120 → *100 instead.
Start practicing

Practice by lesson

Mixed practice: every lesson

Next up

Easy set 1

11 questions

Start practicing
All easy sets
Next chapter · Chapter 14Trigonometry

Next set