GMAT Quantitative Reasoning

Help Questions

GMAT Quantitative › GMAT Quantitative Reasoning

Questions 1 - 10

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

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

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

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

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

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

Explanation

Examine the figure below, which shows the arc and chord in question.

Circle x

If we extend the figure to depict the circle as the composite of four quarter-circles, each a $90^{\circ }$ arc, we see that is also the side of an inscribed square. A diagonal of this square, which measures $\sqrt{2}$ times this sidelength, or

$(x+2) \sqrt{2} = x\sqrt{2} + 2\sqrt{2}$ ,

is a diameter of this circle. The circumference is $\pi$ times the diameter, or

$\pi ( x\sqrt{2} + 2\sqrt{2}) = x \pi \sqrt{2} + 2 \pi \sqrt{2}$ .

Since a $90^{\circ }$ arc is one fourth of a circle, the length of arc $\widehat{AB}$ is

$\frac{1}{4}( x \pi \sqrt{2} + 2 \pi \sqrt{2})$

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

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

Two circles are constructed; one is inscribed inside a given equilateral triangle, and the other is circumscribed about the same triangle.

The circumscribed circle has circumference $20 \pi$ . Give the area of the inscribed circle.

$25 \pi$

$400 \pi$

$50 \pi$

$100 \pi$

$200 \pi$

Explanation

Examine the diagram below, which shows the triangle, its three altitudes, and the two circles.

Thingy

The three altitudes of an equilateral triangle divide one another into two segments each, the longer of which is twice the length of the shorter. The length of each of the longer segments is the radius of the circumscribed circle, and the length of each of the shorter segments is the radius of the inscribed circle. Therefore, the inscribed circle has half the radius of the circumscribed circle.

The circumscribed circle has circumference $20 \pi$ , so its radius is

$20 \pi \div 2 \pi = 10$

The inscribed circle has radius half this, or 5, so its area is

$A= \pi r^{2} = \pi \cdot 5^{2}= 25 \pi$

Two circles are constructed; one is inscribed inside a given equilateral triangle, and the other is circumscribed about the same triangle.

The inscribed circle has circumference $20 \pi$ . Give the area of the circumscribed circle.

$400 \pi$

$200 \pi$

$100 \pi$

$25 \pi$

$50 \pi$

Explanation

Examine the diagram below, which shows the triangle, its three altitudes, and the two circles.

Thingy

The three altitudes of an equilateral triangle divide one another into two segments each, the longer of which is twice the length of the shorter. The length of each of the longer segments is the radius of the circumscribed circle, and the length of each of the shorter segments is the radius of the inscribed circle. Therefore, the circumscribed circle has twice the radius of the inscribed circle.

The inscribed circle has circumference $20 \pi$ , so its radius is

$20 \pi \div 2 \pi = 10$

The circumscribed circle has radius twice this, or 20, so its area is

$A= \pi r^{2} = \pi \cdot 20^{2}= 400 \pi$

What percentage of a circle is a sector if the angle of the sector is $27^{\circ}$ ?

Explanation

The full measure of a circle is $360^{\circ}$ , so any sector will cover whatever fraction of the circle that its angle is of $360^{\circ}$ . We are given a sector with an angle of $27^{\circ}$ , so this sector will cover a percentage of the circle equal to whatever fraction $27^{\circ}$ is of $360^{\circ}$ . This gives us:

$\frac{27^{\circ}}{360^{\circ}}=\frac{3}{40}=0.075=7.5%$

Sector

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

Statement 1: Arc $\widehat{AB}$ has length $30 \pi$ .

Statement 2: Arc $\widehat{BD}$ has length $60 \pi$ .

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.

EITHER statement ALONE is 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.

Explanation

Assume Statement 1 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle $\widehat{AB}$ is, and therefore, the central angle of the sector cannot be determined. Also, no information about the circle can be determined. A similar argument can be given for Statement 2 being insufficient.

Now assume both statements are true. Then the length of semicircle $\widehat{ABD}$ is equal to $30 \pi + 60 \pi = 90 \pi$ . The circumference is twice this, or $180 \pi$ . The radius can be calculated as $r = C \div( 2 \pi )=180 \pi \div (2 \pi) = 90$ , and the area, $\pi r^{2}= \pi \cdot 90^{2} = 8,100 \pi$ . Also, $\widehat{AB}$ is $\frac{30 \pi}{180 \pi}= \frac{1}{6}$ of the circle, and the area of the sector can now be calculated as $\frac{1}{6} \times 8,100 \pi = 1,350 \pi$ .

A given sector covers of a circle. What is the corresponding angle of the sector?

$252^{\circ}$

$300^{\circ}$

$222^{\circ}$

$202^{\circ}$

$272^{\circ}$

Explanation

A circle comprises $360^\circ$ , so a sector comprising of the circle will have an angle that is of .

Therefore:

$\angle=0.70(360)=\frac{7}{10}(360)=252^{\circ}$

What is the arc length for a sector with a central angle of $45^{\circ}$ if the radius of the circle is ?

$\frac{3\pi }{4}$

$\frac{\pi }{4}$

$\frac{3\pi }{2}$

$\frac{\pi }{2}$

$\pi$

Explanation

Using the formula for arc length, we can plug in the given angle and radius to calculate the length of the arc that subtends the central angle of the sector. The angle, however, must be in radians, so we make sure to convert degrees accordingly by multiplying the given angle by $\frac{\pi }{180}$ :

$S=r\theta$

$S=3(45^{\circ})(\frac{\pi }{180^{\circ}})=3(\frac{\pi }{4})=\frac{3\pi}{4}$

A triangle has 2 sides length 5 and 12. Which of the following could be the perimeter of the triangle?

I. 20

II. 25

III. 30

II and III only.

I only

III only

I and II only

All 3 are possible.

Explanation

For a triangle, the sum of the two shortest sides must be greater than that of the longest. We are given two sides as 5 and 12. Our third side must be greater than 7, since if it were smaller than that we would have where is the unknown side. It must also be smaller than 17 since were it larger, we would have .

Thus our perimeter will be between and . Only II and III are in this range.

Two angles of an isosceles triangle measure $(x + 20) ^{\circ}$ and $\left ( 100-x\right )^{ \circ}$ . What are the possible values of ?

$x = 40 \textrm{ or }x = 80$

$x = 60 \textrm{ or } x = 80$

Explanation

In an isosceles triangle, at least two angles measure the same. Therefore, one of three things happens:

Case 1: The two given angles have the same measure.

$2x \div 2 =80 \div 2$

The angle measures are $60 ^{\circ}, 60 ^{\circ}, 60 ^{\circ}$ , making the triangle equianglular and, subsequently, equilateral. An equilateral triangle is considered isosceles, so this is a possible scenario.

Case 2: The third angle has measure $(x + 20) ^{\circ}$ .

Then, since the sum of the angle measures is 180,

$\left (x + 20 \right ) +\left (x +20 \right )+ \left ( 100-x \right ) = 180$

as before

Case 3: The third angle has measure $\left ( 100-x\right )^{ \circ}$

$\left (x + 20 \right ) +\left (100-x \right )+ \left ( 100-x \right ) = 180$

as before.

Thus, the only possible value of is 40.

$\Delta ABC$ is an isosceles triangle with perimeter 43; the length of each of its sides can be given by a prime whole number. What is the greatest possible length of its longest side?

This triangle cannot exist.

Explanation

We are looking for ways to add three primes to yield a sum of 43. Two or all three (since an equilateral triangle is considered isosceles) must be equal (although, since 43 is not a multiple of three, only two can be equal).

We will set the shared sidelength of the congruent sides to each prime number in turn up to 19:

By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate the first three. , , and include numbers that are not prime (21, 15, 9). This leaves us with only one possibility:

- greatest length 19

19 is the correct choice.

Page 1 of 344

Return to subject

AI TutorYour personal study buddy

Question of the DayDaily practice to build your skills

FlashcardsStudy and memorize key concepts

AI SolverStep-by-step problem solutions

Learn by ConceptMaster concepts step by step

Practice TestsTest your skills with exam questions

QuizzesTest your knowledge with a quiz

GamesLearn through free games