Radius - GMAT Quantitative

Card 0 of 304

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

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

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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:

Compare your answer with the correct one above

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 $5 \sqrt{ 2}$ .

Compare your answer with the correct one above

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

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

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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 $3 \sqrt{2 }$

Compare your answer with the correct one above

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:

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

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

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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 .

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

Question

Give the radius of a circle on the coordinate plane.

Statement 1: A right triangle with a hypotenuse with endpoints and can be inscribed in the circle.

Statement 2: A right triangle with a leg with endpoints and can be inscribed in the circle.

Answer

If a right triangle can be inscribed inside a given circle, then its hypotenuse has a length equal to the diameter of the circle, and the radius of the circle can be calculated as half this. Statement 1 gives sufficient information to find this, since the length of the hypotenuse is the distance between its endpoints and , which is ; the diameter of the circle is 20, and the radius is half this, or 10. From Statement 2, we can only find the length of one leg of an inscribed right triangle, so the length of the hypotenuse is still open to question.

Compare your answer with the correct one above

Question

Give the radius of a circle on the coordinate plane.

Statement 1: A square whose vertices include and can be inscribed inside the circle.

Statement 2: A right triangle whose vertices include and can be inscribed inside the circle.

Answer

Assume Statement 1 alone. The length of a segment with the given endpoints can be calculated using the distance formula. However, it is not clear whether the points are opposite vertices, in which case the segment is a diagonal of the square, or the points are consecutive vertices, in which case the segment is a side of the square, making the diagonal of the square $\sqrt{2 }$ times this length. The length of the diagonal of the inscribed square cannot be determined for certain; since the diameter of the circle is equal to the length of the diagonal, the diameter cannot be determined, and since the radius is half this, the radius cannot be determined.

Assume Statement 2 alone. The length of a segment with the endpoints can be calculated using the distance formula. However, it is not clear whether the segment is a hypotenuse of the triangle or not; the diameter of a circle is equal to the length of the hypotenuse of an inscribed right triangle, so knowing this is necessary.

Assume both statements to be true. The two statements together give four points of the circle; since three points uniquely define a circle, the circle can be located; subsequently, the radius can be found.

Compare your answer with the correct one above

Question

Give the radius of the above circle with center .

Statement 1: $\angle AOB$ is a $60 ^{\circ }$ angle.

Statement 2: $\angle ACB$ is a $30 ^{\circ }$ angle.

Answer

The two statements together only give information about angle measures; arc degree measures can be deduced from this information but not any arc lengths or side lengths. Without this information, we cannot obtain the radius of this circle.

Compare your answer with the correct one above

Question

Between Circle 1 and Circle 2, which has the greater radius?

Statement 1: Circle 1 has as a diameter $\overline{AB}$ ; Circle 2 has as a diameter $\overline{AC}$ ; $\bigtriangleup ABC$ is a right triangle.

Statement 2: $\overline{AB}$ and $\overline{AC}$ are the legs of $\bigtriangleup ABC$ .

Answer

Assume both statements are true. Since we have no way to determine which, if either, of the legs of $\bigtriangleup ABC$ is the longer, we have no way to compare the diameters, and, consequently, no way to compare the radii, of the circles with those segments as diameters.

Compare your answer with the correct one above

Tap the card to reveal the answer