ISEE Upper Level Quantitative : Circles

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

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Example Questions

Example Question #1 : Circles

The clock in the classroom reads 5:00pm. What is the angle that the hands are forming?

Possible Answers:

\displaystyle 50^{\circ}

\displaystyle 150^{\circ}

\displaystyle 360^{\circ}

\displaystyle 180^{\circ}

\displaystyle 50^{\circ}

Correct answer:

\displaystyle 150^{\circ}

Explanation:

Since the clock is a circle, you can determine that the total number of degrees inside the circle is 360. Since a clock has 12 numbers, we can divide 360 by 12 to see what the angle is between two numbers that are right next to each other. Thus, we can see that the angle between two numbers right next to each other is \displaystyle 30^{\circ}. However, the clock is reading 5:00, so there are five numbers we have to take in to account. Therefore, we multiply 30 by 5, which gives us \displaystyle 150^{\circ} as our answer.

Example Question #1 : Circles

The time on a clock reads 5:00. What is the measure of the central angle formed by the hands of the clock?

Possible Answers:

\displaystyle 120^{\circ}

\displaystyle 150^{\circ}

\displaystyle 50^{\circ}

\displaystyle 240^{\circ}

\displaystyle 180^{\circ}

Correct answer:

\displaystyle 150^{\circ}

Explanation:

First, remember that the number of degrees in a circle is 360. Then, figure out how many degrees are in between each number on the face of the clock. Since there are 12 numbers, there are \displaystyle 30^{\circ} between each number. Since the time reads 5:00, multiply \displaystyle 30\cdot 5, which yields \displaystyle 150^{\circ}.

Example Question #133 : Geometry

Chords 1

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

(a) \displaystyle ab

(b) 3 

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, 

\displaystyle a \cdot b = 1.5 \cdot 1.5

or

\displaystyle ab = 2.25

Therefore, \displaystyle ab < 3.

Example Question #1 : Circles

Secant

Figure NOT drawn to scale

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

Which is the greater quantity?

(a) \displaystyle NO

(b) 25

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:

(a) and (b) are equal

Explanation:

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

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

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

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

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

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

Example Question #3 : Circles

Chords 1

Figure NOT drawn to scale.

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

(a) \displaystyle t

(b) 7

Possible Answers:

(a) is the greater quantity

(a) and (b) are equal

(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 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, 

\displaystyle t \cdot 4t = 8 \cdot 24

Solving for \displaystyle t:

\displaystyle 4t ^{2} = 192

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

\displaystyle t ^{2} = 48

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

Example Question #134 : Plane Geometry

 

Secant

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

Which is the greater quantity?

(a) \displaystyle NB

(b) 32

Possible Answers:

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

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,

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

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

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

Simplifying, then solving for \displaystyle t:

\displaystyle t \cdot (2t) = 256

\displaystyle 2t ^{2} = 256

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

\displaystyle t ^{2} = 128

 

To compare \displaystyle NB to 32, it suffices to compare their squares: 

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

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

\displaystyle 32^{2} = 1,024

\displaystyle 512 < 1,024, so

\displaystyle (NB)^{2} < 32 ^{2}

it follows that

\displaystyle NB < 32.

Example Question #4 : Circles

Secant

Figure NOT drawn to scale

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

Which is the greater quantity?

(a) \displaystyle t

(b) 8

Possible Answers:

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

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

Correct answer:

(a) and (b) are equal

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 intersected by the secant; in other words,

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

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

\displaystyle (2t) ^{2} = t (t + 24)

Simplifying and solving for \displaystyle t:

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

\displaystyle 4 t^{2} = t ^{2} + 24t

\displaystyle 4 t^{2} - t ^{2} - 24t = t ^{2} + 24t - t ^{2} - 24t

\displaystyle 3 t^{2}- 24t = 0

Factoring out \displaystyle 3t:

\displaystyle 3 t (t-8) = 0

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

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

Example Question #2 : Circles

Chords 1

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

(a) \displaystyle \frac{a}{d}

(b) \displaystyle \frac{c}{b}

Possible Answers:

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

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, 

\displaystyle ab = cd

Divide both sides of this equation by \displaystyle bd, then cancelling:

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

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

The two quantities are equal.

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

The area of Circle B is four times that of Circle A. The area of Circle C is four times that of Circle B. Which is the greater quantity?

(a) Twice the radius of Circle B

(b) The sum of the radius of Circle A and the radius of Circle C

Possible Answers:

It cannot be determined from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

Let \displaystyle r be the radius of Circle A. Then its area is \displaystyle \pi r^{2}.

The area of Circle B is \displaystyle 4\pi r^{2} =2^{2}\pi r^{2} = \pi (2r)^{2}, so the radius of Circle B is twice that of Circle A; by a similar argument, the radius of Circle C is twice that of Circle B, or \displaystyle 2 (2r ) = 4r.

(a) Twice the radius of circle B is \displaystyle 2 (2r) = 4r.

(b) The sum of the radii of Circles A and B is \displaystyle r + 4r = 5r.

This makes (b) greater.

Example Question #2 : Circles

The time is now 1:45 PM. Since noon, the tip of the minute hand of a large clock has moved \displaystyle \frac{14\pi}{3} feet. How long is the minute hand of the clock?

Possible Answers:

\displaystyle 1 \textrm{ ft }3 \textrm{ in }

\displaystyle 1 \textrm{ ft }4 \textrm{ in } 

\displaystyle 2 \textrm{ ft }8 \textrm{ in }

\displaystyle 2 \textrm{ ft }

\displaystyle 2 \textrm{ ft }6 \textrm{ in }

Correct answer:

\displaystyle 1 \textrm{ ft }4 \textrm{ in } 

Explanation:

Every hour, the tip of the minute hand travels the circumference of a circle. Between noon and 1:45 PM, one and three-fourths hours pass, so the tip travels \displaystyle 1 \frac{3}{4} or \displaystyle \frac{7}{4} times this circumference. The length of the minute hand is the radius of this circle \displaystyle r, and the circumference of the circle is \displaystyle C = 2 \pi r, so the distance the tip travels is \displaystyle \frac{7}{4} this, or

\displaystyle \frac{7}{4} C = \frac{7}{4} \cdot 2 \pi r = \frac{7 \pi r }{2}

Set this equal to \displaystyle \frac{14\pi}{3} feet:

\displaystyle \frac{7 \pi r }{2} = \frac{14\pi}{3}

\displaystyle \frac{7 \pi r }{2} \times \frac{2}{7 \pi}= \frac{14\pi}{3}\times \frac{2}{7 \pi}

\displaystyle r= \frac{2}{3}\times \frac{2}{1} = \frac{4}{3} = 1 \frac{1}{3} feet.

This is equivalent to 1 foot 4 inches.

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