Geometry

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GMAT Quantitative › Geometry

Questions 1 - 10

The radius of Circle A is equal to the sidelength of Square B. A sector of Circle A has the same area as Square B. Which of the following is the degree measure of this sector?

$\left (\frac{360}{\pi} \right )^{\circ }$

$\left (\frac{180}{\pi} \right )^{\circ }$

$\left (\frac{90}{\pi} \right )^{\circ }$

$\left (\frac{120}{\pi} \right )^{\circ }$

$\left (\frac{60}{\pi} \right )^{\circ }$

Explanation

The radius of Circle A and the length of a side of the square are the same - we will call each . The area of the circle is $A_{a}= \pi r^{2}$ ; that of the square is $A_{b}= r^{2}$ . Therefore, a sector of the circle with area $r^{2}$ will be $\frac{1}{\pi}$ of the circle, which is a sector of measure

$\frac{1}{\pi} \times 360 ^{\circ } = \left (\frac{360}{\pi} \right )^{\circ }$

Sector

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1: $m \angle ACB = 30 ^{\circ }$

Statement 2: $m \angle BOD = 120 ^{\circ }$

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice this measure, or $60 ^{\circ }$ .

Statement 2 alone asserts that $m \angle BOD = 120 ^{\circ }$ . By angle addition, $m\angle AOB = 180 ^{\circ } - m \angle BOD = 180 ^{\circ } - 120 ^{\circ } = 60 ^{\circ }$ .

Either statement alone tells us that the shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, and that the white sector is $\frac{5}{6}$ of it; it can be subsequently calculated that the ratio of the areas is $\frac{5}{6} : \frac{1}{6}$ , or .

What is the circumference of Circle ?

1.) The diameter of the circle is .

2.) The area of the circle is $25\pi$ .

Each statement alone is sufficient to solve the question.

Statement 1 is sufficient to solve the question, but Statement 2 is not sufficient to solve the question.

Both statements taken together are sufficient to solve the question.

Statement 2 is sufficient to solve the question, but Statement 1 is not sufficient to solve the question.

Neither statement is sufficient to solve the question. More information is needed.

Explanation

We are asked to find the circumference of Circle and are given the diameter and the area. We also know that $C=2\pi r$ . Taking each statement individually:

1.) The diameter is and we know that the radius $r=\frac{d}{2}$ , so $C=2\pi r=2\pi\left(\frac{d}{2}\right)=\pi d=10\pi$ . Therefore, Statement 1 is sufficient to solve for the circumference of the circle by itself.

2.) The area of Circle is $A=\pi r^{2}=25\pi$ , so we can determine that the radius . Since the circumference $C=2\pi r$ , Statement 2 is is sufficient to solve for the circumference of the circle by itself.

Sector

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1:

Statement 2: $m \angle ACB = 30 ^{\circ }$

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

We are asking for the ratio of the areas of the sectors, not the actual areas. The answer is the same regardless of the actual area of the circle, so information about linear measurements such as radius, diameter, and circumference is useless. Statement 2 alone is unhelpful.

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . $\angle ACB$ is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice its measure, or $2 \times 30^{\circ }= 60 ^{\circ }$ . From angle addition, this can be subtracted from $180^{\circ }$ to yield the measure of central angle $\angle BOD$ of the shaded sector, which is $120^{\circ }$ . That makes that sector $\frac{120^{\circ }}{360^{\circ }} = \frac{1}{3}$ of the circle. The white sector is $\frac{2}{3}$ of the circle, and the ratio of the areas can be determined to be $\frac{2}{3} : \frac{1}{3}$ , or .

Ron is making a box in the shape of a cube. He needs to know how much wood he needs. Find the surface area of the box.

I) The diagonal distance across the box will be equivalent to $4\sqrt{3}ft$ .

II) Half the length of one side is .

Either statement is sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Both statements are needed to answer the question.

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Explanation

To find the surface area of a cube, we need the length of one side.

Statement I gives the diagonal, we can use this to find the length of one side.

Statement II gives us a clue about the length of one side; we can use that to find the full length of one side.

The following formula gives us the surface area of a cube:

Use Statement I to find the length of the side with the following formula, where is the diagonal and is the side length:

$d=\sqrt{3}a$

$4\sqrt{3}=\sqrt{3}a$

So, using Statement I, we find the surface area to be

$SA=6\cdot 4^2=6\cdot 16=84 ft^2$

Using Statement, we get that the length of one side is two times two:

$2\cdot 2=4ft$

Again, use the surface area formula to get the following:

$SA=6\cdot 4^2=6\cdot 16=84 ft^2$

What is the length of the edge of a cube?

Its volume is 1,728 cubic meters.
Its surface area is 864 square meters

EACH statement ALONE is sufficient.

Statement 1 ALONE is sufficient, but Statement 2 alone is not sufficient.

Statement 2 ALONE is sufficient, but Statement 1 alone is not sufficient.

Statements 1 and 2 TOGETHER are not sufficient.

BOTH statements TOGETHER are sufficient, but neither statement ALONE is sufficient.

Explanation

Call the sidelength, surface area, and volume of the cube , , and , respectively.

Then

$V = s^{3}$

or, equivalently,

$s = \sqrt[3]{V}$

So, given statement 1 alone - that is, given only the volume, you can demonstrate the sidelength to be

$s = \sqrt[3]{1,728} = 12$

Also,

$A = 6s^{2}$

or, equivalently,

$s = \sqrt{\frac{A}{6}}$

Given statement 2 alone - that is, given only the surface area, you can demonstrate the sidelength to be

$s = \sqrt{\frac{864}{6}} = \sqrt{144} = 12$

Therefore, the answer is that either statement alone is sufficient.

What is the equation of a given circle on the coordinate plane?

Statement 1: Its center is at the origin.

Statement 2: One of its diameters has endpoints and .

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

You cannot determine the equation of a circle knowing only the center, as in Statement 1.

But given the coordinates of any diameter of a circle, as in Statement 2, you can use the midpoint formula to find the center, and the distance formula to find the distance from the center to either endpoint - this is the radius. Once you know the center and the radius, just apply them to the standard form of the equation of a circle.

Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: $m \angle BCD = 120^{\circ }$ .

Statement 2: The circle has circumference $28 \pi$ .

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle $\angle BOD$ of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield radius $28 \pi \div 2 \pi = 14$ , and that can be substituted for in the formula $A = \pi r^{2}$ to find the area: $A = \pi \cdot 14^{2} = 196 \pi$ .

However, it provides no clue that might yield $m \angle BOD$ .

From Statement 2 alone, we can find $m \angle BOD$ . $\angle BCD$ , an inscribed angle, intercepts an arc twice its measure - this arc is $\widehat{BAD}$ , which has measure $240^{\circ }$ . $\widehat{B D}$ , the corresponding minor arc, will have measure $360 ^{\circ}- m \widehat{BAD} = 360 ^{\circ} - 240 ^{\circ} = 120 ^{\circ}$ . This gives us $m \angle BOD$ , but no clue that yields the area.

Now assume both statements are true. The area is $196 \pi$ and the shaded sector is $\frac{120^{\circ }}{360^{\circ }}= \frac{1}{3}$ of the circle, so the area can be calculated to be $\frac{1}{3} \times 196 \pi = \frac{196 \pi}{3}$ .

Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: The sector with central angle $\angle BOD$ has area $48 \pi$ .

Statement 2: $\widehat{BD }= 8 \pi$ .

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

Assume Statement 1 alone. No clues are given about the measure of $\angle BOD$ , so that of $\angle AOB$ , and, subsequently, the area of the shaded sector, cannot be determined.

Assume Statement 2 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle $\widehat{BD }$ , or, subsequently, $\widehat{AB}$ , is, and therefore, the central angle of the sector cannot be determined. Also, no information about the area of the circle can be determined.

Now assume both statements are true. Let be the radius of the circle and $N ^{\circ }$ be the measure of $\angle BOD$ . Then:

$\frac{N}{360} \cdot \pi r ^{2} = 48 \pi$

and

$\frac{N}{360} \cdot 2 \pi r = 8 \pi$

The statements can be simplified as

$r ^{2}N = 17,280$

and

From these two statements:

$\frac{r ^{2}N }{rN}=\frac{ 17,280}{1,440}$

; the second statement can be solved for :

$m \angle BOD = 120^{\circ }$ , so $m \angle AOB = 60^{\circ }$ .

Since , the circle has area $A = \pi r^{2} = \pi \cdot 12^{2} = 144 \pi$ . Since we know the central angle of the shaded sector as well as the area of the circle, we can calculate the area of the sector as

$\frac{1}{6} \times 144 \pi = 24\pi$ .

$\angle 1$ is an exterior angle of $\bigtriangleup ABC$ at $\angle A$ .

Is $\bigtriangleup ABC$ an acute triangle, a right triangle, or an obtuse triangle?

Statement 1: $\angle 1$ is an acute angle.

Statement 2: $m \angle B + m \angle C = 80 ^{\circ }$

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Explanation

Exterior angle $\angle 1$ forms a linear pair with its interior angle $\angle B AC$ . Either both are right, or one is acute and one is obtuse. From Statement 1 alone, since $\angle 1$ is acute, $\angle B AC$ is obtuse, and $\bigtriangleup ABC$ is an obtuse triangle.

Statement 2 alone also provides sufficient information; the sum of the measures of interior angles of a triangle is $180 ^{\circ }$ ; since the sum of the measures of two of them, $m \angle B$ and $m \angle C$ , is $80 ^{\circ }$ , the other angle, $\angle B AC$ , has measure $180 ^{\circ } - 80 ^{\circ } = 100 ^{\circ } > 90 ^{\circ }$ , making $\angle B AC$ obtuse, and making $\bigtriangleup ABC$ an obtuse triangle.

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