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ISEE Upper Level Quantitative : Plane Geometry

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 #5 : 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:

(a) is the greater quantity

(a) and (b) are equal

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

(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,

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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Example Question #1 : 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:

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It cannot be determined from the information given.

Correct answer:

(b) is greater.

Explanation:

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

The area of Circle B is $4\pi r^{2} =2^{2}\pi r^{2} = \pi (2r)^{2}$ , 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 2 (2r ) = 4r .

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

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

This makes (b) greater.

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Example Question #2 : How To Find The Length Of A Radius

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

Possible Answers:

$2 \textrm{ ft }$

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

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

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

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

Correct answer:

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

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 $1 \frac{3}{4}$ or $\frac{7}{4}$ times this circumference. The length of the minute hand is the radius of this circle , and the circumference of the circle is $C = 2 \pi r$ , so the distance the tip travels is $\frac{7}{4}$ this, or

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

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

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

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

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

This is equivalent to 1 foot 4 inches.

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Example Question #2 : Radius

The tip of the minute hand of a giant clock has traveled $40 \pi$ feet since noon. It is now 2:30 PM. Which is the greater quantity?

(A) The length of the minute hand

(B) Three yards

Possible Answers:

(A) and (B) are equal

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

(A) is greater

(B) is greater

Correct answer:

(B) is greater

Explanation:

Betwen noon and 2:30 PM, the minute hand has made two and one-half revolutions; that is, the tip of minute hand has traveled the circumference of its circle two and one-half times. Therefore,

$C \cdot 2 \frac{1}{2} = 40 \pi$

$C = 40 \pi \div 2 \frac{1}{2} = \frac{40 \pi}{1} \div \frac{5}{2} = \frac{40 \pi}{1} \cdot \frac{2}{5}= \frac{80 \pi}{5} = 16\pi$ feet.

The radius of this circle is the length of the minute hand. We can use the circumference formula to find this:

$2 \pi r = C$

$2 \pi r = 16 \pi$

$r = 16 \pi \div 2 \pi = 8$

The minute hand is eight feet long, which is less than three yards (nine feet), so (B0 is greater.

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Example Question #141 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Compare the two quantities:

Quantity A: The radius of a circle with area of $64\pi$

Quantity B: The radius of a circle with circumference of $50\pi$

Possible Answers:

The relationship cannot be determined from the information given.

The quantity in Column A is greater.

The two quantities are equal.

The quantity in Column B is greater.

Correct answer:

The quantity in Column B is greater.

Explanation:

Recall for this question that the formulae for the area and circumference of a circle are, respectively:

$A=\pi * r^2$

$C=2*\pi*r$

For our two quantities, we have:

Quantity A

$64\pi = \pi * r^2$

Therefore, r^2 = 64

Taking the square root of both sides, we get: r = 8

Quantity B

$50\pi = 2 * \pi * r$

Therefore, r = 25

Therefore, quantity B is greater.

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

Compare the two quantities:

Quantity A: The radius of a circle with area of $144\pi$

Quantity B: The radius of a circle with circumference of $24\pi$

Possible Answers:

The quantity in Column B is greater.

The relationship cannot be determined from the information given.

The two quantities are equal.

The quantity in Column A is greater.

Correct answer:

The two quantities are equal.

Explanation:

Recall for this question that the formulae for the area and circumference of a circle are, respectively:

$A=\pi * r^2$

$C=2*\pi*r$

For our two quantities, we have:

Quantity A

$144\pi = \pi * r^2$

Therefore, r^2 = 144

Taking the square root of both sides, we get: r = 12

Quantity B

$24\pi = 2 * \pi * r$

Therefore, r = 12

Therefore, the two quantities are equal.

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

If the diameter of a circle is equal to $5x^{2}+4x-2$ , then what is the value of the radius?

Possible Answers:

$2.5x^{2}+2x-1$

$2.5x^{2}+2x+1$

$2(x^{2}+x)-1$

7x-1

Correct answer:

$2.5x^{2}+2x-1$

Explanation:

Given that the radius is equal to half the diameter, the value of the radius would be equal to $5x^{2}+4x-2$ divided by 2. This gives us:

$\frac{5x^{2}+4x-2}{2}$

$2.5x^{2}+2x-1$

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Example Question #3 : Radius

The area of a circle is $81x^{2}$ . Give its radius in terms of .

(Assume is positive.)

Possible Answers:

$\frac{81x^{2}}{\pi}$

$9 \pi x$

$\frac{9x}{\pi}$

$\frac{9x}{\sqrt{\pi}}$

$\frac{81x^{2}}{2\pi}$

Correct answer:

$\frac{9x}{\sqrt{\pi}}$

Explanation:

The relation between the area of a circle and its radius is given by the formula

$A = \pi r^{2}$

Since

$A= 81x^{2}$ :

$81x^{2} = \pi r^{2}$

We solve for :

$\pi r^{2} = 81x^{2}$

$\frac{\pi r^{2} }{\pi}= \frac{81x^{2}}{\pi}$

$r^{2} = \frac{81x^{2}}{\pi}$

$\sqrt{r^{2} }= \sqrt{\frac{81x^{2}}{\pi}}$

Since is positive, as is :

$r =\frac{ \sqrt{81} \cdot \sqrt{x^{2}}}{\sqrt{\pi}} = \frac{9x}{\sqrt{\pi}}$

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

The areas of five circles form an arithmetic sequence. The smallest circle has radius 4; the second smallest circle has radius 8. Give the radius of the largest circle.

Possible Answers:

$4 \sqrt{13}$

$8\sqrt{13}$

Correct answer:

$4 \sqrt{13}$

Explanation:

The area of a circle with radius is $A = \pi r^{2}$ . Therefore, the areas of the circles with radii 4 and 8, respectively, are

$A_{1} = \pi \cdot 4^{2} = 16 \pi$

and

$A_{2} = \pi \cdot 8^{2} = 64\pi$

The areas form an arithmetic sequence, the common difference of which is

$64 \pi - 16 \pi = 48 \pi$ .

The circles will have areas:

$16 \pi, 64 \pi, 112 \pi, 160 \pi , 208 \pi$

Since the area of the largest circle is $208 \pi$ , we can find the radius as follows:

The radius can be calculated now:

$\pi r^{2} = 208 \pi$

$r^{2} = 208$

$r = \sqrt{208} = \sqrt{16} \cdot \sqrt{13} = 4 \sqrt{13}$

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Example Question #1 : How To Find The Area Of A Circle

Circle B has a radius $\frac{2}{3}$ as long as that of Circle A.

Which is the greater quantity?

(a) The area of Circle A

(b) Twice the area of Circle B

Possible Answers:

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

Correct answer:

(a) is greater.

Explanation:

If we call the radius of Circle A , then the radius of Circle B is $\frac{2}{3} r$ .

The areas of the circles are:

(a) $\pi r^{2}$

(b) $\pi \left( \frac{2}{3} r \right ) ^{2} = \frac{4}{9} \pi r^{2}$

Twice the area of Circle B is

$2 \cdot \pi \left( \frac{2}{3} r \right ) ^{2} =2 \cdot \frac{4}{9} \pi r^{2} = \frac{8}{9} \pi r^{2} < \pi r^{2}$ ,

making (a) the greater number.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Plane Geometry

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #5 : How To Find The Length Of A Chord

Example Question #1 : Radius

Example Question #2 : How To Find The Length Of A Radius

Example Question #2 : Radius

Example Question #141 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

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

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

Example Question #3 : Radius

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

Example Question #1 : How To Find The Area Of A Circle

All ISEE Upper Level Quantitative Resources

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