### All GMAT Math Resources

## Example Questions

### Example Question #6 : Circles

The equation of a given circle is

.

What is the radius of the circle?

Statement 1:

Statement 2: The circle passes through the origin.

**Possible Answers:**

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.

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

**Correct answer:**

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

The standard form of the equation of a circle is

,

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 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 . This is the radius.

### Example Question #7 : Circles

Find the radius of circle B

I) Circle B has a circumference of .

II) Circle B has an area of .

**Possible Answers:**

Both statements taken together are sufficient to solve the question.

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

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

Each statement alone is enough to solve the question.

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

**Correct answer:**

Each statement alone is enough to solve the question.

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

Given the following equations:

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

### Example Question #8 : Circles

Calculate the length of the radius of a circle.

Statement 1): The circumference of the circle is .

Statement 2):

**Possible Answers:**

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

BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question.

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

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

EACH statement ALONE is sufficient.

**Correct answer:**

EACH statement ALONE is sufficient.

Statement 1) gives the circumference of a circle. The formula for finding the circumference of a circle is . The radius can be solved by using this formula.

Statement 2) gives the standard form of a circle, where 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.

### Example Question #9 : Circles

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.

**Possible Answers:**

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

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.

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

Either statement is sufficient to answer the question.

**Correct answer:**

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

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.

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

### Example Question #10 : Circles

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

Statement 1: A right triangle, both of whose legs measures 10, can be inscribed inside Circle 1; a square of perimeter 48 can be inscribed inside Circle 2.

Statement 2: A square of area 100 can be inscribed inside Circle 1; a 30-60-90 triangle, one of whose legs measures , can be inscribed in side Circle 2.

**Possible Answers:**

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.

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.

**Correct answer:**

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

The length of a diagonal of a square inscribed inside a circle is equal to the diameter of the circle, and half this is the radius. If a right triangle is inscribed inside a circle, The hypotenuse of a right triangle inscribed inside a circle is also a diameter, and half the length is the radius. Since each statement alone mentions an inscribed square in one circle to an inscribed right triangle in the other, we need only compare the length of a diagonal of the former to that of the hypotenuse of the latter.

Assume Statement 1 alone. The square inscribed inside Circle 2 has perimeter 48; its sidelength is one fourth this, or 12, and the length of a diagonal is times this, or . The hypotenuse of the right triangle inscribed in Circle 1, with both legs equal to 10 - which is a 45-45-90 triangle, being isosceles - is, by the 45-45-90 Theorem, times the length of a leg, or . The diagonal of the square inscribed inside Circle 2 is longer than the hypotenuse of the triangle inscribed inside Circle 1, so Circle 2 has the greater radius.

Assume Statement 2 alone. A square of area 100 - and, subsequently, of sidelength the square root of this, or 10 - can be inscribed inside Circle 1; its diagonal will have length this, or .

The 30-60-90 triangle inscribed inside Circle 2 has a leg of length . However, Statement 2 does not make it clear whether the leg is the shorter leg or the longer leg. If it is the shorter leg, then by the 30-60-90 Theorem, the hypotenuse is twice , or ; if it is the longer leg, then by the 30-60-90 Theorem, the hypotenuse is times , or

.

In the first scenario, since , Circle 2 has the longer radius. In the second, since (this can be seen by noting that and ), Circle 1 has the greater radius.

### Example Question #1 : Dsq: Calculating The Length Of A Radius

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.

**Possible Answers:**

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.

**Correct 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 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.

### Example Question #2 : Dsq: Calculating The Length Of A Radius

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

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

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.

### Example Question #3 : Dsq: Calculating The Length Of A Radius

An equilateral triangle is inscribed inside a circle; is the midpoint of . What is the radius of the circle?

Statement 1: has area .

Statement 2: .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

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

Since is equilateral, , , and are all altitudes that insersect at the center of the circumscribed circle, , so that . 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

, or, equivalently,

Once is calculated, then, since is also a perpendicular bisector of and a bisector of , making a 30-60-90 triangle, can be calculated to be one half of ; can be multiplied by 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 to obtain radius .

Statement 2 gives us explicitly, so we can take two thirds of this to get the radius

.

### Example Question #4 : Dsq: Calculating The Length Of A Radius

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

Statement 1: A arc of Circle 1 has length equal to one fourth the circumference of Circle 2.

Statement 2: A sector of Circle 2 has area equal to four ninths that of Circle 1.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

Assume Statement 1 alone. A arc of Circle 1 has the circumference of Circle 1. Since this is also the circumference of Circle 2, then, if we let be the circumferences,

,

and

.

This gives Circle 2 the greater circumference and, subsequently, the greater radius.

Assume Statement 2 alone. A sector of Circle 2 has area of Circle 2. Since its area is also equal to that of Circle 1, then, if are the areas of Circle 1 and Circle 2, respectively, then

,

and

This gives Circle 2 the greater area and, subsequently, the greater radius.

### Example Question #5 : Dsq: Calculating The Length Of A Radius

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.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct 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.