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SAT Math : How to find the length of a radius

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Example Question #361 : Plane Geometry

Chords

In the above diagram, $\overarc {AD}$ and $\overarc {BC}$ have lengths $9 \pi$ and $18\pi$ , respectively. Give the radius of the circle.

Possible Answers:

135

The question cannot be answered with the information given.

$33\frac{3}{4}$

$67\frac{1}{2}$

$16\frac{7}{8}$

Correct answer:

$33\frac{3}{4}$

Explanation:

If two chords of a circle intersect, the measure of the angles they form is equal to half the sum of the measures of the angles they intercept that is,

$\frac{1}{2} \left ( m \overarc {AD}+ m \overarc {BC} \right ) = m \angle AXD$

The ratio of the measures of arcs on the same circle is equal to that of their lengths, so

$\frac{ m \overarc {BC} }{m \overarc {AD} } = \frac{18 \pi}{9 \pi} = 2$

and

$m \overarc {BC} = 2 \cdot m \overarc {AD}$

Substituting:

$\frac{1}{2} \left ( m \overarc {AD}+ 2 \cdot m \overarc {AD} \right ) =72^{\circ }$

$\frac{1}{2} \left ( 3 \cdot m \overarc {AD} \right ) =72^{\circ }$

$\frac{3}{2} \cdot m \overarc {AD} =72^{\circ }$

$\frac{2} {3}\cdot \frac{3}{2} \cdot m \overarc {AD} =\frac{2} {3}\cdot 72^{\circ }$

$m \overarc {AD} =48^{\circ }$

if is the circumference of the circle, and the length of the arc $\overarc {AD}$ is ,

$l = \frac{m \overarc{AC}}{360 ^{\circ }} \cdot C$

$\overarc {AD}$ has length $9 \pi$ and measure $48^{\circ }$ so

$9 \pi = \frac{48^{\circ }}{360 ^{\circ }} \cdot C$

$9 \pi = \frac{2 }{15 } \cdot C$

Since, if the radius is , $C = 2 \pi r$ ,

$9 \pi = \frac{2 }{15 } \cdot 2 \pi r$

$9 \pi = \frac{4 \pi }{15 } r$

Solve for :

$\frac{4 \pi }{15 } r = 9 \pi$

$\frac{15 } {4 \pi } \cdot \frac{4 \pi }{15 } r =\frac{15 } {4 \pi } \cdot 9 \pi$

$r =\frac{135} {4 } = 33\frac{3}{4}$

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Example Question #362 : Plane Geometry

Secant

In the above diagram, $\overarc {AT}$ and $\overarc {BT}$ have lengths $20 \pi$ and $75 \pi$ , respectively. Give the radius of the circle.

Possible Answers:

The question cannot be answered with the information given.

242

198

121

Correct answer:

Explanation:

The ratio of the degree measures of the arcs is the same as that of their lengths. Therefore,

$\frac{m \overarc {BT}}{m \overarc {AT}} = \frac{75 \pi}{20 \pi} = \frac{15}{4}$

and

$m \overarc {BT}= \frac{15}{4} \cdot m \overarc {AT}$

If a secant and a tangent are drawn to a circle from a point outside the circle, the measure of the angle they form is equal to half the difference of the measures of their intercepted arcs; therefore,

$\frac{1}{2} \left ( m\overarc {BT} - m \overarc {AT} \right ) = m \angle BNT$

Substituting:

$\frac{1}{2} \left ( m\overarc {BT} - m \overarc {AT} \right ) = 50 ^{\circ }$

$\frac{1}{2} \left ( \frac{15}{4} \cdot m \overarc {AT} - m \overarc {AT} \right ) = 50 ^{\circ }$

$\frac{11}{8} \cdot m \overarc {AT} = 50 ^{\circ }$

$\frac{8}{11} \cdot \frac{11}{8} \cdot m \overarc {AT} =\frac{8}{11} \cdot 50 ^{\circ }$

$m \overarc {AT} = 36 \frac{4}{11}^{\circ }$

Since $\overarc {AT}$ has length $20 \pi$ , then, if we let be the circumference of the circle,

$\frac{C}{360^{\circ }} = \frac{20 \pi} {36\frac{4}{11}^{\circ } }$

$\frac{C}{360^{\circ }} \cdot 360^{\circ }= \frac{20 \pi} {36\frac{4}{11}^{\circ } }\cdot 360^{\circ }$

$C=198 \pi$

Divide the circumference by $2 \pi$ to obtain the radius:

$r = \frac{C}{2 \pi} = \frac{198 \pi }{2 \pi} = 99$

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Example Question #61 : Radius

Give the radius of a circle with diameter fifteen yards.

Possible Answers:

$300 \textrm{ in}$

$270 \textrm{ in}$

$280 \textrm{ in}$

$240 \textrm{ in}$

$320 \textrm{ in}$

Correct answer:

$270 \textrm{ in}$

Explanation:

Convert fifteen yards to inches by multiplying by 36:

$15 \textrm{ yd} \times 36 \textrm{ in/yd} = 540 \textrm{ in}$

The radius of a circle is one half its diameter, so multiply this by $\frac{1}{2}$ :

$\frac{1}{2} \times 540 \textrm{ in} = 270 \textrm{ in}$

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Example Question #61 : Radius

Tangents

In the above diagram, $\overarc {TU}$ has length $30 \pi$ . Give the radius of the circle to the nearest whole number.

Possible Answers:

The question cannot be answered with the information given.

102

Correct answer:

Explanation:

Call $t = m \overarc {TU}$ . The measure of the corresponding major arc is $360 ^{\circ } - m \overarc {TU}= 360 ^{\circ } -t$

If two tangents are drawn to a circle from a point outside the circle, the measure of the angle they form is equal to half the difference of the measures of their intercepted arcs; therefore

$\frac{1}{2} \left (360 ^{\circ } -t -t \right ) = m \angle BNT$

Substituting:

$\frac{1}{2} \left (360 ^{\circ } -2t\right ) = 66 ^{\circ }$

$180 ^{\circ } -t = 66 ^{\circ }$

$180 ^{\circ } -t + t - 66 ^{\circ } = 66 ^{\circ } + t - 66 ^{\circ }$

$114 ^{\circ } = t$

Therefore, $m \overarc {TU} = 114 ^{\circ }$ . Since $\overarc {TU}$ has length $30 \pi$ , it follows that if is the circumference of the circle,

$\frac{C}{360^{\circ }} = \frac{30 \pi }{m \overarc{TU}}$

$\frac{C}{360^{\circ }} = \frac{30 \pi }{114^{\circ }}$

$C= \frac{30 \pi }{114^{\circ }} \cdot 360^{\circ }$

$C = 94\frac{14}{19} \pi$

Divide the circumference by $2 \pi$ to obtain the radius:

$r = \frac{C}{2 \pi} = \frac{94 \frac{14}{19}\pi }{2 \pi} = 47 \frac{7}{19}$ .

This makes 47 the correct choice.

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SAT Math : How to find the length of a radius

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #361 : Plane Geometry

Example Question #362 : Plane Geometry

Example Question #61 : Radius

Example Question #61 : Radius

All SAT Math Resources

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