GMAT Math : GMAT Quantitative Reasoning

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Example Questions

Example Question #1 : How To Graph A Function

The chord of a $60^{\circ }$ central angle of a circle with area $40 \pi$ has what length?

Possible Answers:

$10\sqrt{2}$

$4 \sqrt{5}$

$2 \sqrt{10}$

Correct answer:

$2 \sqrt{10}$

Explanation:

The radius of a circle with area $40 \pi$ can be found as follows:

$\pi r^{2} = A$

$\pi r^{2} = 40 \pi$

$r^{2} = 40$

$r = \sqrt{40 } = \sqrt{4} \cdot \sqrt{10} = 2 \sqrt{10}$

The circle, the central angle, and the chord are shown below:

Chord

By way of the Isosceles Triangle Theorem, $\Delta AOB$ can be proved equilateral, so $AB = AO= 2 \sqrt{10}$ , the correct response.

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Example Question #341 : Gmat Quantitative Reasoning

The chord of a $120^{\circ }$ central angle of a circle with area $32 \pi$ has what length?

Possible Answers:

$4 \sqrt{6}$

$4\sqrt{3}$

$4 \sqrt{2}$

Correct answer:

$4 \sqrt{6}$

Explanation:

The radius of a circle with area $32 \pi$ can be found as follows:

$\pi r^{2} = A$

$\pi r^{2} = 32 \pi$

$r^{2} = 32$

$r = \sqrt{32} = \sqrt{16} \cdot \sqrt{2} = 4 \sqrt{2}$

The circle, the central angle, and the chord are shown below, along with $\overline{AM}$ , which bisects isosceles $\Delta AOB$

Chord

We concentrate on $\Delta AOM$ , a 30-60-90 triangle. By the 30-60-90 Theorem,

$OM = \frac{1}{2}AO = \frac{1}{2} \cdot 4 \sqrt{2} = 2\sqrt{2}$

and

$AM = OM \sqrt{3} = 2 \sqrt{2} \cdot \sqrt{3} = 2 \sqrt{6}$

The chord $\overline{AB}$ has length twice this, or

$2 \cdot 2 \sqrt{6} = 4 \sqrt{6}$

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Example Question #2 : Calculating The Length Of A Chord

The chord of a $120^{\circ }$ central angle of a circle with circumference $40 \pi$ has what length?

Possible Answers:

$20 \sqrt{3}$

$20 \sqrt{6}$

$20 \sqrt{2}$

Correct answer:

$20 \sqrt{3}$

Explanation:

A circle with circumference $40 \pi$ has as its radius

$40 \div 2\pi = 20$ .

The circle, the central angle, and the chord are shown below, along with $\overline{AM}$ , which bisects isosceles $\Delta AOB$ :

Chord

We concentrate on $\Delta AOM$ , a 30-60-90 triangle. By the 30-60-90 Theorem,

$\overline{OM}$ has half the length of $\overline{AO}$ , so OM = 10

and

$AM = OM \sqrt{3} =10 \sqrt{3}$

The chord $\overline{AB}$ has length twice this, or

$2 \cdot 10 \sqrt{3}= 20 \sqrt{3}$

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Example Question #342 : Gmat Quantitative Reasoning

The chord of a $90^{\circ }$ central angle of a circle with area $50 \pi$ has what length?

Possible Answers:

$5\sqrt{3}$

$5\sqrt{2}$

$10 \sqrt{2}$

Correct answer:

Explanation:

The radius of a circle with area $50 \pi$ can be found as follows:

$\pi r^{2} = A$

$\pi r^{2} = 50 \pi$

$r^{2} = 50$

$r = \sqrt{50 } = \sqrt{25} \cdot \sqrt{2} = 5 \sqrt{2}$

The circle, the central angle, and the chord are shown below:

Chord

By way of the Isosceles Triangle Theorem, $\Delta AOB$ can be proved a 45-45-90 triangle with legs of length $5 \sqrt{2}$ . Its hypotenuse has length $\sqrt{ 2}$ times this, or

$5 \sqrt{2} \times \sqrt{2}= 5 \times 2 = 10$

This is the correct response.

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Example Question #3 : Calculating The Length Of A Chord

The chord of a $90^{\circ }$ central angle of a circle with circumference $60 \pi$ has what length?

Possible Answers:

$30\sqrt{3}$

$60\sqrt{2}$

$30\sqrt{2}$

Correct answer:

$30\sqrt{2}$

Explanation:

A circle with circumference $60 \pi$ has as its radius

$60\pi \div 2\pi = 30$ .

The circle, the central angle, and the chord are shown below:

Chord

By way of the Isosceles Triangle Theorem, $\Delta AOB$ can be proved a 45-45-90 triangle with legs of length 30. By the 45-45-90 Theorem, its hypotenuse - the chord of the central angle - has length $\sqrt{ 2}$ times this, or $30 \sqrt{2}$ . This is the correct response.

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Example Question #5 : Calculating The Length Of A Chord

Consider the Circle :

Circle3

(Figure not drawn to scale.)

If $\angle BOX$ is a $90^{\circ}$ angle, what is the measure of segment ?

Possible Answers:

$30 \:m$

$15\sqrt{3} \:m$

$15 \:m$

$\large \large 15\sqrt{2} \:m$

$30\sqrt{2} \:m$

Correct answer:

$\large \large 15\sqrt{2} \:m$

Explanation:

This is a triangle question in disguise. We have a ninety-degree triangle with two sides made up of the radii of the circle. This means the other two angles ( $\angle OXB$ and $\angle OBX$ ) must be $45^{\circ}$ each.

Use the 45/45/90 triangle ratios to find the final side. Additionally, you could use Pythagorean Theorem to find the missing side.

45/45/90 side length ratios:

$x/x/\sqrt{2}x$

$x=radius=15\:m$

Segment $BX=15\sqrt{2}\:m$

Or, using the Pythagorean Theorem, a^2+b^2=c^2 by rearranging it and solving for , the hypotenuse, which in this case is segment :

$BX=\sqrt{15^2+15^2}=\sqrt{450}=\sqrt{225*2}=\sqrt{225}*\sqrt{2}=15*\sqrt{2}=15\sqrt2\:m$

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Example Question #4 : Calculating The Length Of A Chord

Calculate the length of a chord in a circle with a radius of , given that the perpendicular distance from the center to the chord is .

Possible Answers:

Correct answer:

Explanation:

We are given the radius of the circle and the perpendicular distance from its center to the chord, which is all we need to calculate the length of the chord. Using the formula for chord length that involves these two quantities, we find the solution as follows, where is the chord length, is the perpendicular distance from the center of the circle to the chord, and is the radius:

$L=2\sqrt{r^2-d^2}=2\sqrt{5^2-3^3}=2\sqrt{25-9}=2\sqrt{16}=8$

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Example Question #2 : How To Graph A Function

The chord of a $60^{\circ }$ central angle of a circle with circumference $90 \pi$ has what length?

Possible Answers:

$45\sqrt{2}$

$45\sqrt{3}$

$90\sqrt{2}$

Correct answer:

Explanation:

A circle with circumference $90 \pi$ has as its radius

$90\pi \div 2\pi = 45$ .

The circle, the central angle, and the chord are shown below:

Chord

By way of the Isosceles Triangle Theorem, $\Delta AOB$ can be proved equilateral, so AB = AO= 45 , the correct response.

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Example Question #9 : Calculating The Length Of A Chord

The arc $\widehat{AB}$ of a circle measures $90^{\circ }$ and has length x + 2 . Give the length of the chord $\overline{AB}$ .

Possible Answers:

$\frac{ \pi \sqrt{2}}{4} x + \frac{ \pi \sqrt{2}}{2}$

$\frac{2 \sqrt{2}}{\pi}x + \frac{4 \sqrt{2}}{\pi}$

$\frac{ \sqrt{2}}{\pi}x + \frac{ \sqrt{2}}{\pi}$

$\frac{ \pi \sqrt{2}}{2} x + \pi \sqrt{2}$

$\pi x \sqrt{2}+ 2 \pi \sqrt{2}$

Correct answer:

$\frac{2 \sqrt{2}}{\pi}x + \frac{4 \sqrt{2}}{\pi}$

Explanation:

The figure referenced is below.

Circle x

The arc is $\frac{90}{360} = \frac{1}{4}$ of the circle, so the circumference of the circle is

4 (x+2) = 4x+8 .

The radius is this circumference divided by $2 \pi$ , or

$\frac{ 4x+8}{2 \pi} = \frac{2}{\pi}x + \frac{4}{\pi}$ .

$\overline{AB}$ is, consequently, the hypotenuse of an isosceles right triangle with leg length $\frac{2}{\pi}x + \frac{4}{\pi}$ ; by the 45-45-90 Triangle Theorem, its length is $\sqrt{2}$ times this, or

$\left (\frac{2}{\pi}x + \frac{4}{\pi} \right ) \sqrt{2} = \frac{2 \sqrt{2}}{\pi}x + \frac{4 \sqrt{2}}{\pi}$

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Example Question #1 : Rectangles

Rectangles

Note: Figure NOT drawn to scale.

Refer to the above figure. $\textup{Rect }ABCD \sim \textup{Rect }WXYZ$ .

$\frac{AD}{WZ} = \frac{7}{5}$ .

Give the ratio of the area of the shaded region to the area of $\textup{Rect }WXYZ$ .

Possible Answers:

7:5

1:1

5:7

25:24

24:25

Correct answer:

24:25

Explanation:

The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of $\textup{Rect }ABCD$ to that of $\textup{Rect }WXYZ$ is

$\left ( \frac{7}{5} \right )^{2} = \frac{49}{25}$ .

So if the area of $\textup{Rect }WXYZ$ is , the area of $\textup{Rect }ABCD$ is $\frac{49}{25} N$

The area of the shaded region is the difference between the areas of the rectangles, making this area

$\frac{49}{25} N - N = \frac{24}{25} N$ .

The desired ratio is 24 to 25.

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GMAT Math : GMAT Quantitative Reasoning

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : How To Graph A Function

Example Question #341 : Gmat Quantitative Reasoning

Example Question #2 : Calculating The Length Of A Chord

Example Question #342 : Gmat Quantitative Reasoning

Example Question #3 : Calculating The Length Of A Chord

Example Question #5 : Calculating The Length Of A Chord

Example Question #4 : Calculating The Length Of A Chord

Example Question #2 : How To Graph A Function

Example Question #9 : Calculating The Length Of A Chord

Example Question #1 : Rectangles

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