GMAT Quantitative - Circles

The arc $\widehat{AB}$ of a circle measures $60^{\circ }$ . The chord of the arc, $\overline{AB}$ , has length . Give the length of the radius of the circle.

$2 \pi x + 4 \pi$

$\pi x +2 \pi$

$\textup{None of the other responses gives a correct answer.}$

Explanation

A circle can be divided into congruent arcs that measure

$\frac{1}{6} \cdot 360 ^{\circ } = 60 ^{\circ }$ .

If the (congruent) chords are constructed, the figure will be a regular hexagon. The radius of this hexagon will be equal to the length of one side - one $60 ^{\circ }$ chord of the circle; this radius will coincide with the radius of the circle. Therefore, the radius of the circle is the length of chord $\overline{AB}$ , or .

A $60^{\circ }$ arc of a circle measures . Give the radius of this circle.

$\frac{3}{\pi}x+\frac{12}{\pi}$

$\frac{6}{\pi}x+\frac{24}{\pi}$

$6 \pi x+24 \pi$

$9 \pi x^{2}+ 72 \pi x+ 144 \pi$

$3\pi x+12\pi$

Explanation

A $60^{\circ }$ arc of a circle is $\frac{60}{360 } = \frac{1}{6}$ of the circle. Since the length of this arc is , the circumference is $\textup{six times }$ this, or

The radius of a circle is its circumference divided by $2 \pi$ ; therefore, the radius is

$r = \frac{C}{2 \pi} = \frac{6x+24}{2\pi} = \frac{3}{\pi}x+\frac{12}{\pi}$

What is the area of a circle with a diameter of ?

$324\pi cm^2$

$81\pi cm^2$

$54\pi cm^2$

$36\pi cm^2$

$90\pi cm^2$

Explanation

The area of a circle is defined by $A=\pi r^{2}$ , where is the radius of the circle. We are provided with the diameter of the circle, which is twice the length of .

If , then $r=\frac{d}{2}=\frac{18cm}{2}=9cm$

Then, solving for :

$A=\pi r^{2}$

$A=\pi (9cm)^{2}$

$A=81\pi cm^2$

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

$2 \sqrt{10}$

$10\sqrt{2}$

$4 \sqrt{5}$

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.

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

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

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

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

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

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

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

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

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

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 .

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:

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

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 circle has an area of $81\pi$ . What is the length of its diameter?

Not enough information provided

Explanation

The area of a circle is defined by the equation $A=\pi r^{2}$ , where is the length of the circle's radius. The radius, in turn, is defined by the equation $r=\frac{d}{2}$ , where is the length of the circle's diameter.

Given $A=81\pi$ , we can deduce that $r^{2}=81$ and therefore . Then, since $r=\frac{d}{2}$ , .

The points and form a line which passes through the center of circle Q. Both points are on circle Q.

To the nearest hundreth, what is the length of the radius of circle Q?

Explanation

To begin this problem, we need to recognize that the distance between points L and K is our diameter. Segment LK passes from one point on circle Q through the center, to another point on circle Q. Sounds like a diameter to me! Use distance formula to find the length of LK.

$d=\sqrt{(x-x')^2+(y-y')^2}$

Plug in our points and simplify:

$d=\sqrt{(9--4)^2+(3-5)^2}=\sqrt{13^2+2^2}=\sqrt{173}\approx13.15$

Now, don't be fooled into choosing 13.15. That is our diameter, so our radius will be half of 13.15, or 6.575. This rounds to 6.58

Circles

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