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ISEE Upper Level Quantitative : How to find the length of a chord

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

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All ISEE Upper Level Quantitative Resources

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Example Questions

Example Question #1 : Circles

Chords 1

Refer to the above figure. Which is the greater quantity?

(a)

(b) 3

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

It is impossible to determine which is greater from the information given

(a) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

If two chords intersect inside a circle, both chords are cut in a way such that the products of the lengths of the two chords formed in each are the same - in other words,

$a \cdot b = 1.5 \cdot 1.5$

ab = 2.25

Therefore, ab <3 .

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Example Question #1 : Circles

Secant

Figure NOT drawn to scale

In the above figure, is the center of the circle, and $\overline{NT}$ is a tangent to the circle. Also, the circumference of the circle is $14 \pi$ .

Which is the greater quantity?

(a)

(b) 25

Possible Answers:

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

$\overline{OT}$ is a radius of the circle from the center to the point of tangency of $\overline{NT}$ , so

$\overline{OT} \perp \overline{NT}$ ,

and $\bigtriangleup NTO$ is a right triangle. The length of leg $\overline{NT}$ is known to be 24. The other leg $\overline{OT}$ is a radius radius; we can find its length by dividing the circumference by $2 \pi$ :

$OT = r = \frac{C}{2\pi} = \frac{14 \pi}{2\pi} = 7$

The length hypotenuse, $\overline{NO}$ , can be found by applying the Pythagorean Theorem:

$NO = \sqrt{(NT)^{2} + (OT)^{2}}= \sqrt{24^{2} + 7^{2}}=\sqrt{576 + 49} = \sqrt{625} = 25$ .

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Example Question #1 : Chords

Chords 1

Figure NOT drawn to scale.

Refer to the above figure. Which is the greater quantity?

(a)

(b) 7

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

If two chords intersect inside a circle, both chords are cut in a way such that the products of the lengths of the two chords formed in each are the same - in other words,

$t \cdot 4t = 8 \cdot 24$

Solving for :

$4t ^{2} = 192$

$4t ^{2} \div 4 = 192 \div 4$

$t ^{2} = 48$

Since $t ^{2} < 49$ , it follows that $t < \sqrt{49 }$ , or t < 7 .

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Example Question #4 : How To Find The Length Of A Chord

Secant

In the above figure, $\overline{NT}$ is a tangent to the circle.

Which is the greater quantity?

(a)

(b) 32

Possible Answers:

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

It is impossible to determine which is greater from the information given

Correct answer:

(b) is the greater quantity

Explanation:

If a secant segment and a tangent segment are constructed to a circle from a point outside it, the square of the distance to the circle along the tangent is equal to the product of the distances to the two points on the circle along the secant; in other words,

$NA \cdot NB = (NT )^{2}$

$NA \cdot (NA + AB)= (NT )^{2}$

$t \cdot (t+t) = 16 ^{2}$

Simplifying, then solving for :

$t \cdot (2t) = 256$

$2t ^{2} = 256$

$2t ^{2} \div 2 = 256 \div 2$

$t ^{2} = 128$

To compare to 32, it suffices to compare their squares:

NB = NA + AB= t + t = 2t , so, applying the Power of a Product Principle, then substituting,

$(NB)^{2} = (2t) ^{2} = 2^{2} \cdot t^{2} = 4 t ^{2} = 4 \cdot 128 = 512$

$32^{2} = 1,024$

512 < 1,024 , so

$(NB)^{2} < 32 ^{2}$ ;

it follows that

NB < 32 .

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Example Question #1 : How To Find The Length Of A Chord

Secant

Figure NOT drawn to scale

In the above figure, $\overline{NT}$ is a tangent to the circle.

Which is the greater quantity?

(a)

(b) 8

Possible Answers:

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(a) is the greater quantity

Correct answer:

(a) and (b) are equal

Explanation:

$(NT)^{2}= NA \cdot NB$

$(NT)^{2}= NA \cdot (NA+AB)$

$(2t) ^{2} = t (t + 24)$

Simplifying and solving for :

$2^{2} \cdot t^{2} = t \cdot t + t \cdot 24$

$4 t^{2} = t ^{2} + 24t$

$4 t^{2} - t ^{2} - 24t = t ^{2} + 24t - t ^{2} - 24t$

$3 t^{2}- 24t = 0$

Factoring out :

3 t (t-8) = 0

Either t = 0 - which is impossible, since must be positive, or

t- 8 =0 , in which case t = 8 .

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Example Question #6 : How To Find The Length Of A Chord

Chords 1

Refer to the above figure. Which is the greater quantity?

(a) $\frac{a}{d}$

(b) $\frac{c}{b}$

Possible Answers:

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

If two chords intersect inside a circle, both chords are cut in a way such that the products of the lengths of the two chords formed in each are the same - in other words,

ab = cd

Divide both sides of this equation by , then cancelling:

$\frac{ab }{bd}= \frac{cd}{bd}$

$\frac{a }{d}= \frac{c}{b}$

The two quantities are equal.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : How to find the length of a chord

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : Circles

Example Question #1 : Circles

Example Question #1 : Chords

Example Question #4 : How To Find The Length Of A Chord

Example Question #1 : How To Find The Length Of A Chord

Example Question #6 : How To Find The Length Of A Chord

All ISEE Upper Level Quantitative Resources

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