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ISEE Upper Level Math : Circles

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

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Example Questions

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Example Question #5 : 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 #6 : 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 = 66

t = 78

t = 72

t = 60

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 #7 : 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 =63

t =72

t = 81

t= 54

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 : Geometry

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

Possible Answers:

$37.5\%$

$45\%$

$12.5 \%$

$125\%$

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 #2 : How To Find The Percentage Of A Sector From An Angle

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:

$15\%$

$18.06\%$

$65\%$

$81.94\%$

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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ISEE Upper Level Math : Circles

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #5 : How To Find The Angle Of A Sector

Example Question #6 : How To Find The Angle Of A Sector

Example Question #7 : How To Find The Angle Of A Sector

Example Question #221 : Geometry

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

All ISEE Upper Level Math Resources

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