DSQ: Calculating the length of a radius - GMAT Quantitative Reasoning

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Question

Right triangle $\bigtriangleup ABC$ is inscribed inside a circle. What is the radius of the circle?

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

Statement 2:

Answer

If a right triangle is inscribed inside a circle, the hypotenuse of the triangle is a diameter of the circle; therefore, its length is the diameter, and half this is the radius.

Statement 1 alone does not give the hypotenuse of the triangle, or, for that matter, any of the sidelengths. Statement 2 alone gives one sidelength, but does not state whether it is the hypotenuse or not.

Assume both statements are true. Since $m \angle C = 30 ^{\circ }$ in right triangle $\bigtriangleup ABC$ , then either $m \angle A = 90^{\circ }$ and $m \angle B = 60^{\circ }$ , or vice versa. In either event, $\overline{AB }$ , being opposite the $30 ^{\circ }$ angle, is the short leg of a 30-60-90 triangle, and, by the 30-60-90 Theorem, the hypotenuse is twice its length. This is twice 18, or 36. This is the diameter of the circle, and the radius is half this, or 18.

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Question

The equation of a given circle is

$(x-A) ^{2} + (y - B) ^{2} = C ^{2}$ .

What is the radius of the circle?

Statement 1:

Statement 2: The circle passes through the origin.

Answer

The standard form of the equation of a circle is

$(x-h) ^{2} + (y - k) ^{2} =r ^{2}$ ,

where the radius is and the center is .

The equation given is this same form, with replacing , replacing , and replacing , so to find the radius, we need to find .

Statement 1 alone tells us that the center is $\left (5,5 \right )$ but it tells us nothing about the radius. Statement 2 alone tells us only that the circle passes through .

The two together, however, reveal enough information to give the radius. The radius is the distance from the center to a point on the circle, so we can use the distance formula to find the distance between and $\left (5,5 \right )$ . This is the radius.

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Question

Find the radius of circle B

I) Circle B has a circumference of $\small 8\pi m$ .

II) Circle B has an area of $\small 16\pi m^2$ .

Answer

We are given the area and circumference of a circle and asked to find the radius.

Given the following equations:

$\small a=\pi r^2$

$\small c=2\pi r$

We can use either equation to work backwards and find our radius, therefore; Each statement alone is enough to solve the question.

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Question

Let circle represent the base of a lamp. Find its radius.

I) The ratio of circle diameter to circumference is approximately .

II) The base of the lamp will cover an area of square inches.

Answer

To find the radius of a cricle, we need its circumference, diameter, or area.

I) Seems to be helpful, but it is really just giving us pi, so it is not sufficient.

II) Gives us the area of the circle, which we can use to work backward to find the radius.

$A=\pi r^2$

$12.5=\pi r^2 \rightarrow \sqrt{\frac{12.5}{\pi}}=r$

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

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Question

Calculate the length of the radius of a circle.

Statement 1): The circumference of the circle is .

Statement 2):

Answer

Statement 1) gives the circumference of a circle. The formula for finding the circumference of a circle is $C=2\pi r$ . The radius can be solved by using this formula.

Statement 2) gives the standard form of a circle, where $\left ( h,k\right )$ is the center of the circle:

The radius is also given in the equation.

Therefore, either statement alone is enough to solve for the radius of a circle.

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Question

An equilateral triangle $\bigtriangleup ABC$ is inscribed inside a circle; is the midpoint of $\overline{AB}$ . What is the radius of the circle?

Statement 1: $\bigtriangleup ABC$ has area $36 \sqrt{3}$ .

Statement 2: $CM = 6 \sqrt{3}$ .

Answer

First, locate the other midpoints of the sides of the triangle and construct the segments from each vertex to the opposite midpoint.

Since $\bigtriangleup ABC$ is equilateral, $\overline{AP}$ , $\overline{BN}$ , and $\overline{CM}$ are all altitudes that insersect at the center of the circumscribed circle, , so that $CO = \frac{2}{3} \cdot CM$ . is the radius of the circumscribed circle.

Assume Statement 1 alone. The length of one side of an equilateral triangle can be calculated using the formula

$A = \frac{s^{2}\sqrt{3}}{4}$ , or, equivalently,

$\frac{(AB)^{2}\sqrt{3}}{4} = 36 \sqrt{3}$

Once is calculated, then, since $\overline{CM}$ is also a perpendicular bisector of $\overline{AB}$ and a bisector of $\angle ACB$ , making $\bigtriangleup MBC$ a 30-60-90 triangle, can be calculated to be one half of ; can be multiplied by $\sqrt{3}$ to yield , and, since the three altitudes of an equilateral triangle divide one another into segments whose lengths have ratio 2:1, can be multiplied by $\frac{2}{3}$ to obtain radius .

Statement 2 gives us $CM = 6 \sqrt{3}$ explicitly, so we can take two thirds of this to get the radius

$CO = \frac{2}{3} \cdot CM = \frac{2}{3} \cdot 6\sqrt{3}= 4 \sqrt{3}$ .

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Question

Right triangle $\bigtriangleup ABC$ is inscribed inside a circle. What is the radius of the circle?

Statement 1:

Statement 2:

Answer

If a right triangle is inscribed inside a circle, the hypotenuse of the triangle is a diameter of the circle; therefore, its length is the diameter, and half this is the radius. However, from the two statements, it cannot be determined which segment is the hypotenuse.

If $\overline{AB}$ is the hypotenuse, then the radius is half its length; since , the radius is 10.

If $\overline{AC}$ is the hypotenuse, then, since the hypotenuse is the longest side of a right triangle, - that is, . The radius is greater than 10.

Therefore, the radius depends on which side is the hypotenuse; since that is not clear, the radius cannot be determined.

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Question

Rectangle is inscribed inside a circle. What is the radius of the circle?

Statement 1:

Statement 2:

Answer

The diameter of a circle with an inscribed rectangle is equal to the length of the diagonal of the rectangle; once this diameter is found, it can be divided by 2 to yield the radius.

Statement 1 alone gives this length, from which the radius can be found to be $25 \div 2 = 12 \frac{1}{2}$ . Statement 2 alone gives only the length of one set of opposite sides, from which the length of the diagonal cannot be determined.

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Question

A polygon is inscribed inside a circle. What is the radius of the circle?

Statement 1: Each side of the polygon measures 10.

Statement 2: The inscribed polygon is a regular hexagon.

Answer

Statement 1 alone yields insufficient information, since, as seen in the diagram below, the circles that circumscribe a square and a triangle with the same sidelength have different sizes:

Statement 2 is insufficient since it gives no hints about the size of the hexagon.

Assume both statements. The six radii of a regular hexagon divide it into six equilateral triangles, by symmetry; therefore, the radius of a regular hexagon is equal to its sidelength, which is given in Statement 1 as 10. Since the radius of a regular hexagon is equal to that of the circle in which it is inscribed, the circle has radius 10.

This can be seen by examining the figure below:

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Question

Square is inscribed inside a circle. What is the radius of the circle?

Statement 1: Square has area 100.

Statement 2: $SU = 10 \sqrt{2}$ .

Answer

From Statement 2 alone, $\overline{SU}$ , a diagonal of the square, measures $10 \sqrt{ 2}$ . The diameter of the circle is equal to the length of a diagonal of an inscribed square, so the radius of the circle is equal to half this, or $5 \sqrt{ 2}$ .

From Statement 1 alone, since the area of the square is 100, its sidelength is the square root of this, or 10. By the 45-45-90 Theorem, a diagonal of the square measures $\sqrt{2 }$ times this, or $10 \sqrt{ 2}$ , which makes Statement 2 a consequence of Statement 1. Therefore, it follows again that the circle has radius $5 \sqrt{ 2}$ .

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Question

A circle is inscribed inside an equilateral triangle $\bigtriangleup ABC$ . $\overline{A B}$ , $\overline{BC}$ , and $\overline{AC}$ are tangent to the circle at the points , , and , respectively. What is the radius of the circle?

Statement 1: The length of arc $\widehat{XY}$ is $4 \pi$ .

Statement 2: The degree measure of arc $\widehat{XZY}$ is $240 ^{\circ }$ .

Answer

The figure referenced is below:

By symmetry, $\widehat{XY}$ is one third of a circle. Therefore, its length is one third of the circumference, so, if Statement 1 alone is assumed, the circumference can be determined to be $4 \pi \cdot 3 = 12 \pi$ ; this can be divided by $2 \pi$ to yield radius .

Statement 2 yields no helpful information; from the body of the problem, $\widehat{XZY}$ can already be deduced to be two thirds of a circle, or, equivalently, an arc of measure

$\frac{2}{3} \cdot 360 ^{\circ } = 240^{\circ }$ .

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Question

Rectangle is inscribed inside a circle. What is the radius of the circle?

Statement 1: Rectangle has area 200.

Statement 2: Rectangle has perimeter 60.

Answer

The diameter of a circle with an inscribed rectangle is equal to the length of a diagonal of the rectangle , which, given the length and width , can be found using the Pythagorean Theorem:

$d = \sqrt{l^{2}+w^{2}}$

Statement 1 alone gives insufficient information. For example, a 20 by 10 rectangle has area $20 \times 10 = 200$ , and a 40 by 5 rectangle has area $40 \times 5 = 200$ .

The first rectangle has diagonals of length

$d = \sqrt{20^{2}+10^{2}} = \sqrt{400+100} = \sqrt{500}$ .

The second rectangle has diagonals of length

$d = \sqrt{40^{2}+5^{2}} = \sqrt{1,600+25} = \sqrt{1,625}$ ,

Since the diagonals of the rectangles differ, so do the diameters, and, consequently, the radii, of the circles.

Statement 2 alone gives insufficient information for a similar reason. For example, a 20 by 10 rectangle has perimeter , and a 25 by 5 rectangle has perimeter . Again, the first rectangle has diagonals of length $d = \sqrt{500}$ . The second has diagonals of length

$d = \sqrt{25^{2}+5^{2}} = \sqrt{625+25} = \sqrt{650}$ .

Now, assume both statements to be true. We are looking for two numbers whose product is 200 and whose sum is 30 (since the perimeter is twice the sum, or 60). The only such pair of numbers can be found by trial and error to be 20 and 10, so these are the length and width of the rectangle. As shown before, a rectangle with these dimensions has diagonals of length $\sqrt{500} = \sqrt{100} \cdot \sqrt{5 } = 10\sqrt{5 }$ . This is the diameter of the circle in which it is inscribed, so half this, or $5\sqrt{5 }$ , is the radius.

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Question

Right triangle $\bigtriangleup ABC$ is inscribed inside a circle. What is the radius of the circle?

Statement 1: $\bigtriangleup ABC$ has area 36.

Statement 2: $AB = BC = 6\sqrt{2}$

Answer

If a right triangle is inscribed inside a circle, the hypotenuse of the triangle is a diameter of the circle; therefore, its length is the diameter, and half this is the radius.

Assume Statement 1 alone. The area of a right triangle is half the product of the lengths of the legs, which is 36. However, this is not enough to determine the length of the hypotenuse.

For example, if the legs measure 8 and 9, the triangle has area

$A = \frac{1}{2} \cdot 8 \cdot 9 = 36$

By the Pythagorean Theorem, the hypotenuse measures

$\sqrt{8^{2}+9^{2}} = \sqrt{64+81} = \sqrt{145}$ ,

which is the diameter of the circle; half this, or $\frac{ \sqrt{145}}{2}$ , is the radius of the circle.

If the legs measure 6 and 12, the triangle has area

$A = \frac{1}{2} \cdot 6 \cdot 12 = 36$

By the Pythagorean Theorem, the hypotenuse measures

$\sqrt{6^{2}+12^{2}} = \sqrt{36+144} = \sqrt{180}$ ,

which is the diameter of the circle; half this, or $\frac{ \sqrt{180}}{2}$ , is the radius of the circle.

Therefore, Statement 1 is inisufficient to give the radius.

Assume Statement 2 alone. The hyppotenuse of a right triangle must be longer than either leg, so it is impossible for either of the two equally long sides $\overline{AB}$ and $\overline{BC}$ to be the hypotenuse of $\bigtriangleup ABC$ ; they must be its legs. Since the legs are of the same length, $\bigtriangleup ABC$ is a 45-45-90 triangle, and by the 45-45-90 Theorem, hypotenuse $\overline{AC}$ has length $\sqrt{2 }$ that of a leg, or $6\sqrt{2 }$ . This is the diameter of the circle, and the radius is half this, or $3 \sqrt{2 }$

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Question

Note: figure NOT drawn to scale.

Give the radius of the above circle with center .

Statement 1: The shaded sector has area $27 \pi$ .

Statement 2: Arc $\widehat{BD}$ has length $6 \pi$ .

Answer

Let $n^{\circ }$ be the measure of $\angle BOD$ and be radius.

From Statement 1 alone, the area of the shaded sector is

$\frac{n}{360} \cdot \pi r^{2} = 27 \pi$

$\frac{n}{360} \cdot \pi r^{2} \cdot \frac{360 }{n \pi} = 27 \pi \cdot \frac{360 }{n \pi}$

$r^{2} = \frac{9,720}{n}$

$r=\sqrt{ \frac{9,720}{n}}$

However, we have no other information, so we cannot determine the value of the radius.

From Statement 1 alone, the length of the arc of the shaded sector $\widehat{BD}$ is

$\frac{n}{360} \cdot 2 \pi r = 6 \pi$

$\frac{n}{360} \cdot 2 \pi r \cdot \frac{360 }{2 n \pi} = 6 \pi\cdot \frac{360 }{2 n \pi}$

$r = \frac{1,080}{ n }$

Again, we have no other information, so we cannot determine the value of the radius.

Assume both statements hold. From Statements 1 and 2, we have, respectively,

$r^{2} = \frac{9,720}{n}$

and

$r = \frac{1,080}{ n }$

If we divide, we get the radius:

$r^{2} \div r = \frac{9,720}{n} \div \frac{1,080}{ n }$

$r = 9,720 \div 1.080 = 9$ .

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Question

Parallelogram is inscribed inside a circle. What is the radius of the circle?

Statement 1: Each side of Parallelogram has length 20.

Statement 2: $AC = 20 \sqrt{2}$ .

Answer

Opposite angles of a parallelogram are congruent, and if the parallelogram is inscribed, both angles are inscribed as well. Congruent inscribed angles on the same circle intercept congruent arcs; since the two congruent arcs together comprise a circle, each intercepted arc is a semicricle. This makes the angles right angles, and this forces a parallelogram inscribed in a circle to be a rectangle.

Statement 1 alone tells us that this is also a square, and that its sides have length 20. The diagonal of a square, which is also a diameter of the circle that circumscribes it, has length $\sqrt{ 2}$ times that of a side, or $20\sqrt{ 2}$ ; half this, or $10 \sqrt{ 2}$ , is the radius of the circle.

Statement 2 alone gives a diagonal of the rectangle, which, again, is enough to determine the radius of the circle.

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Question

Rectangle is inscribed inside a circle. What is the radius of the circle?

Statement 1:

Statement 2:

Answer

The diameter of a circle that circumscribes a rectangle is equal to the length of a diagonal of the rectangle; the radius is equal to half this.

The two statements together, however, do not yield this. The opposite sides of a rectangle are congruent, so the two statements are actually equivalent; each gives the same dimension of the rectangle. This is insufficient to determine the length of the diagonal.

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Question

Give the radius of a circle on the coordinate plane.

Statement 1: Of the three intercepts of the circle, exactly two are -intercepts, one of which is at the point .

Statement 2: Of the three intercepts of the circle, exactly two are -intercepts, one of which is at the point .

Answer

Statement 1 alone only gives one point through which the circle passes, so no information can be determined about the other points or about the size or location of the circle. A similar argument holds for the insufficiency of Statement 2.

Now assume both statements are true. The circle has exactly three intercepts, but it is given that there are two -intercepts - and one other point - and two -intercepts - and one other point. The unidentified -intercept and the unidentified -intercept must be one and the same, and the only possible way this can happen is for this common point to be the origin . Since three points define a circle, we can now identify the unique circle through the points , , and , and we can figure out its radius.

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Question

The equation of a circle can be written in the form

$\left (x-A \right )^{2}+ \left ( y -B\right )^{2} = C$

Give the radius of the circle of this equation.

Statement 1:

Statement 2:

Answer

The actual form of the equation of a circle is

$\left (x-h \right )^{2}+ \left ( y -k\right )^{2} = r^{2}$ ,

where is the location of the center, and is the radius.

The radius of the circle in the equation

$\left (x-A \right )^{2}+ \left ( y -B\right )^{2} = C$

is therefore $\sqrt{C}$ , making Statement 2 alone sufficient to answer the question - and Statement 1 unhelpful.

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Question

The equation of a circle can be written in the form

$\left (x-A \right )^{2}+ \left ( y -B\right )^{2} = C$

Give the radius of the circle of this equation.

Statement 1:

Statement 2:

Answer

The actual form of the equation of a circle is

$\left (x-h \right )^{2}+ \left ( y -k\right )^{2} = r^{2}$ ,

where is the location of the center, and is the radius.

The radius of the circle in the equation

$\left (x-A \right )^{2}+ \left ( y -B\right )^{2} = C$

is therefore $\sqrt{C}$ . We need to know the value of in the equation.

Assume both statements are true. Then we can add the equations to get :

$\underline{A - B + C = 10}$

But without further information, we cannot determine .

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Question

Give the radius of a circle on the coordinate plane.

Statement 1: The circle has its center at .

Statement 2: The circle has its -intercepts at and .

Answer

Knowing neither the center alone, as given in Statement 1, nor two points alone, as given in Statement 2, is sufficient to find the radius of the circle.

Assume both statements to be true. Knowing the center from Statement 1 and one point on the circle, as given in Statement 2, is enough to determine the radius - use the distance formula to find the distance between the two points and, equivalently, the radius.

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