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ISEE Upper Level Math : Plane Geometry

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Example Questions

Example Question #2 : How To Find The Angle Of A Sector

Inscribed angle

Figure NOT drawn to scale

Refer to the above diagram. $\overarc {ABC}$ is a semicircle. Evaluate $m \overarc {BCA}$ given $\measuredangle B=90^\circ$ .

Possible Answers:

$302^{\circ }$

$292^{\circ }$

$312^{\circ }$

$282^{\circ }$

Correct answer:

$312^{\circ }$

Explanation:

An inscribed angle of a circle that intercepts a semicircle is a right angle; therefore, $\angle B$ , which intercepts the semicircle $\overarc {ABC}$ , is such an angle. Consequently, $\bigtriangleup ABC$ is a right triangle, and $\angle A$ and $\angle C$ are complementary angles. Therefore,

$m \angle A + m \angle C = 90 ^{\circ }$

$y + \left ( \frac{y}{3}+2\right ) = 90$

$y + \frac{1}{3}y +2 = 90$

$\frac{4}{3}y +2 = 90$

$\frac{4}{3}y +2 - 2 = 90 - 2$

$\frac{4}{3}y = 88$

$\frac{3}{4} \cdot \frac{4}{3}y = \frac{3}{4} \cdot 88$

y = 66

$m \angle C = \left (\frac{y}{3}+ 2 \right ) ^{\circ }=\left ( \frac{66}{3}+ 2 \right )^{\circ }=( 22 + 2 )^{\circ }= 24^{\circ }$

Inscribed $\angle C$ intercepts an arc with twice its angle measure; this arc is $\overarc {AB}$ , so

$m \overarc {AB} = 2 \cdot m \angle C = 2 \cdot 24 ^{\circ } = 48 ^{\circ }$ .

The major arc corresponding to this minor arc, $\overarc {BCA}$ , has measure

$m \overarc {BCA} = 360 ^{\circ } - m \overarc {AB} = 360 ^{\circ } - 48^{\circ } = 312^{\circ }$

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Example Question #221 : Geometry

Inscribed angle

Note: Figure NOT drawn to scale

Refer to the above diagram. $\overarc {ABC}$ is a semicircle. Evaluate $m \overarc {AB}$ .

Possible Answers:

$74 ^{\circ }$

$64 ^{\circ }$

$69^{\circ }$

$79 ^{\circ }$

Correct answer:

$74 ^{\circ }$

Explanation:

An inscribed angle of a circle that intercepts a semicircle is a right angle; therefore, $\angle B$ , which intercepts the semicircle $\overarc{ABC}$ , is such an angle. Consequently,

$m \angle B = 90 ^{\circ }$

4x-18 = 90

4x - 18 + 18 = 90 + 18

4x = 108

$4x \div 4 = 108 \div 4$

x = 27

$m \angle C = (x+10)^{\circ } = (27+10)^{\circ }= 37^{\circ }$

Inscribed $\angle C$ intercepts an arc with twice its angle measure; this arc is $\overarc{AB}$ , so

$m \widehat{AB} = 2 \cdot m \angle C = 2 \cdot 37 ^{\circ } = 74 ^{\circ }$ .

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Example Question #1 : How To Find The Angle Of A Sector

Intercepted

In the above diagram, radius AO = 20 .

Calculate the length of $\overarc {AB}$ .

Possible Answers:

$40 \pi$

$8 \pi$

$16 \pi$

$32 \pi$

Correct answer:

$16 \pi$

Explanation:

Inscribed $\angle ACB$ , which measures $72 ^{\circ }$ , intercepts an arc with twice its measure. That arc is $\overarc {AB}$ , which consequently has measure

$72 ^{\circ } \times 2 = 144 ^{\circ }$ .

This makes $\overarc {AB}$ an arc which comprises

$\frac{144}{360} = \frac{144 \div 72}{360 \div 72} = \frac{2}{5}$

of the circle.

The circumference of a circle is $2 \pi$ multiplied by its radius, so

$C = 2 \pi r = 2\pi \cdot 20 = 40 \pi$ .

The length of $\overarc {AB}$ is $\frac{2}{5}$ of this, or $\frac{2}{5} \cdot 40 \pi = 16 \pi$ .

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Example Question #2 : How To Find The Angle Of A Sector

Secant

Figure NOT drawn to scale.

The circumference of the above circle is 120. $\overarc{AB }$ and $\overarc{CD}$ have lengths 10 and 20, respectively. Evaluate .

Possible Answers:

t = 15

t = 24

t = 18

t = 30

Correct answer:

t = 15

Explanation:

The length of $\overarc{AB }$ comprises $\frac{10}{120} = \frac{1}{12}$ of the circumference of the circle. Therefore, its degree measure is $\frac{1}{12} \cdot 360 ^{\circ } = 30^{\circ }$ . Similarly, The length of $\overarc{CD}$ comprises $\frac{20}{120} = \frac{1}{6}$ of the circumference of the circle. Therefore, its degree measure is $\frac{1}{6} \cdot 360 ^{\circ } = 60 ^{\circ }$ .

If two secants are constructed to a circle from an outside point, the degree measure of the angle the secants form is half the difference of those of the arcs intercepted - that is,

$t = \frac{1}{2} ( m \overarc {CD} - m \overarc{AB})$

$t = \frac{1}{2} ( 60 - 30 ) = \frac{1}{2} \cdot 30 = 15$ .

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Example Question #2 : How To Find The Angle Of A Sector

Tangents

Figure NOT drawn to scale.

Refer to the above diagram. $\overarc{AB }$ and $\overarc{ABC}$ have lengths 80 and 160, respectively. Evaluate .

Possible Answers:

t = 72

t = 66

t = 60

t = 78

Correct answer:

t = 60

Explanation:

The circumference of the circle is the sum of the two arc lengths:

80 + 160 = 240

The length of $\overarc{AB }$ comprises $\frac{80}{240} = \frac{1}{3}$ of the circumference of the circle. Therefore, its degree measure is $\frac{1}{3} \cdot 360 ^{\circ } = 120 ^{\circ }$ . Consequently, $\overarc{ACB }$ is an arc of degree measure $360 ^{\circ } - 120 ^{\circ } = 240^{\circ }$ .

The segments shown are both tangents from to the circle. Consequently, the degree measure of the angle they form is half the difference of the angle measures of the arcs they intercept - that is,

$t = \frac{1}{2} ( m \overarc {ACB} - m \overarc{AB})$

$t = \frac{1}{2} ( 240 ^{\circ } - 120^{\circ }) = \frac{1}{2} \cdot 120^{\circ } = 60 ^{\circ }$

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Example Question #3 : How To Find The Angle Of A Sector

Chords

Figure NOT drawn to scale.

The circumference of the above circle is 100. $\overarc{AB }$ and $\overarc{CD}$ have lengths 20 and 15, respectively. Evaluate .

Possible Answers:

t =72

t= 54

t = 81

t =63

Correct answer:

t =63

Explanation:

The length of $\overarc{AB }$ comprises $\frac{20}{100} = \frac{1}{5}$ of the circumference of the circle. Therefore, its degree measure is $\frac{1}{5} \cdot 360 ^{\circ } = 72 ^{\circ }$ . Similarly, The length of $\overarc{CD}$ comprises $\frac{15}{100} = \frac{3}{20}$ of the circumference of the circle. Therefore, its degree measure is $\frac{3}{20} \cdot 360 ^{\circ } = 54 ^{\circ }$ .

If two chords cut each other inside the circle, as $\overline{AC }$ and $\overline{BD}$ do, and one pair of vertical angles are examined, then the degree measure of each angle is half the sum of those of the arcs intercepted - that is,

$t = \frac{1}{2} ( m \overarc{AB}+ m \overarc {CD})$

$t = \frac{1}{2} ( 72 ^{\circ } + 54 ^{\circ })$

$t = \frac{1}{2} ( 126^{\circ })$

$t =63 ^{\circ }$

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Example Question #221 : Plane Geometry

Sector SOW has a central angle of $45^{\circ}$ . What percentage of the circle does it cover?

Possible Answers:

$125\%$

$37.5\%$

$12.5 \%$

$45\%$

Correct answer:

$12.5 \%$

Explanation:

Sector SOW has a central angle of $45^{\circ}$ . What percentage of the circle does it cover?

Recall that there is a total of 360 degrees in a circle. SOW occupies 45 of them. To find the percentage, simply do the following:

$\frac{45^{\circ}}{360^{\circ}}*100=.125*100=12.5\%$

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Example Question #222 : Plane Geometry

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of $65^{\circ}$ . What percentage of the circle is highlighted?

Possible Answers:

$65\%$

$81.94\%$

$18.06\%$

$15\%$

Correct answer:

$18.06\%$

Explanation:

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of $65^{\circ}$ . What percentage of the circle is highlighted?

To find the percentage of a sector, simply put the degree measure of the angle over 360 and multiply by 100.

$Sector=\frac{65^{\circ}}{360^{\circ}}*100\approx 18.06 \%$

So, our answer is 18.06%

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Example Question #1 : How To Find An Angle In Other Quadrilaterals

Three of the interior angles of a quadrilateral measure $51^{\circ }$ , $120 ^{\circ }$ , and $77^{\circ}$ . What is the measure of the fourth interior angle?

Possible Answers:

This quadrilateral cannot exist.

$92^{\circ }$

$122^{\circ }$

$102^{\circ }$

$112^{\circ }$

Correct answer:

$112^{\circ }$

Explanation:

The measures of the angles of a quadrilateral have sum $360^{\circ }$ . If is the measure of the unknown angle, then:

x + 51 + 120 + 77 = 360

x + 248 = 360

x + 248 -248 = 360-248

x = 112

The angle measures $112^{\circ }$ .

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Example Question #1 : How To Find An Angle In Other Quadrilaterals

The angles of a quadrilateral measure $x^{\circ }, x^{\circ }, \left ( x+45 \right ) ^{\circ }, \left ( x+65 \right ) ^{\circ }$ . Evaluate .

Possible Answers:

x = 65

x = 70

x = 62.5

x = 67.5

Correct answer:

x = 62.5

Explanation:

The sum of the degree measures of the angles of a quadrilateral is 360, so we can set up and solve for in the equation:

$x + x + \left ( x+45 \right ) + \left ( x+65 \right ) = 360$

4x+110= 360

4x+110- 110 = 360 - 110

4x = 250

$4x\div 4 = 250 \div 4$

x = 62.5

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ISEE Upper Level Math : Plane Geometry

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #2 : How To Find The Angle Of A Sector

Example Question #221 : Geometry

Example Question #1 : How To Find The Angle Of A Sector

Example Question #2 : How To Find The Angle Of A Sector

Example Question #2 : How To Find The Angle Of A Sector

Example Question #3 : How To Find The Angle Of A Sector

Example Question #221 : Plane Geometry

Example Question #222 : Plane Geometry

Example Question #1 : How To Find An Angle In Other Quadrilaterals

Example Question #1 : How To Find An Angle In Other Quadrilaterals

All ISEE Upper Level Math Resources

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