# GMAT Math : Circles

## Example Questions

### Example Question #12 : Dsq: Calculating The Length Of A Radius

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

Statement 1: Circle 1 has as a diameter ; Circle 2 has as a diameter  is a right triangle.

Statement 2:  is the hypotenuse of .

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.

EITHER 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.

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

Explanation:

Statement 2 is unhelpfful as it gives no information about the circles - only a triangle which is not given in that statement to have any connection to the circles. Statement 1 alone is unhelpful in that it only identifies two sides of a triangle as  diameters of the circles without giving their lengths.

Now assume both statements to be true. The hypotenuse of a right triangle, which Statement 2 gives as , must be the longest side, so  has greater length than . This means that the circle with  as a diameter, Circle 1, must have a greater diameter than one with  as a diameter, Circle 2. Since Circle 1 has the greater diameter, it has the greater radius.

### Example Question #13 : Dsq: Calculating The Length Of A Radius

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.

BOTH statements TOGETHER are insufficient 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 sufficient to answer the question, but NEITHER 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.

Explanation:

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:

### Example Question #14 : Dsq: Calculating The Length Of A Radius

Note: figure NOT drawn to scale.

Give the radius of the above circle with center .

Statement 1: The shaded sector has area .

Statement 2: Arc  has length .

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT 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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

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.

Explanation:

Let  be the measure of  and  be radius.

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

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  is

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,

and

If we divide, we get the radius:

.

### Example Question #15 : Dsq: Calculating The Length Of A Radius

The equation of a circle can be written in the form

Give the radius of the circle of this equation.

Statement 1:

Statement 2:

BOTH statements TOGETHER are insufficient to answer the question.

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:

The actual form of the equation of a circle is

,

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

The radius of the circle in the equation

is therefore . We need to know the value of  in the equation.

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

But without further information, we cannot determine .

### Example Question #16 : Dsq: Calculating The Length Of A Radius

Right triangle  is inscribed inside a circle. What is the radius of the circle?

Statement 1:

Statement 2:

BOTH statements TOGETHER are insufficient to answer the question.

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

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

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, each statement alone gives us the length of only one side, and does not clue us in to which side is the hypotenuse.

Assume both statements are true, however. The hypotenuse must always be the longest side of the right triangle, so if two sides have the same length, they must be the legs. A right triangle with congruent legs is a 45-45-90 triangle, and by the 45-45-90 Theorem, its hypotenuse must measure  the length of a leg. Since each leg of  measures 100, the length of the hypotenuse, and the diameter of the circle, are , and the radius is half this, or .

### Example Question #21 : Dsq: Calculating The Length Of A Radius

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

Statement 1:

Statement 2:

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.

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

Explanation:

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 . Statement 2 alone gives only the length of one set of opposite sides, from which the length of the diagonal cannot be determined.

### Example Question #22 : Dsq: Calculating The Length Of A Radius

Right triangle  is inscribed inside a circle. What is the radius of the circle?

Statement 1:  has area 36.

Statement 2:

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.

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

Explanation:

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

By the Pythagorean Theorem, the hypotenuse measures

,

which is the diameter of the circle; half this, or , is the radius of the circle.

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

By the Pythagorean Theorem, the hypotenuse measures

,

which is the diameter of the circle; half this, or , 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  and  to be the hypotenuse of ; they must be its legs. Since the legs are of the same length,  is a 45-45-90 triangle, and by the 45-45-90 Theorem, hypotenuse  has length  that  of a leg, or . This is the diameter of the circle, and the radius is half this, or

### Example Question #23 : Dsq: Calculating The Length Of A Radius

The equation of a circle can be written in the form

Give the radius of the circle of this equation.

Statement 1:

Statement 2:

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

EITHER 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.

Explanation:

The actual form of the equation of a circle is

,

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

The radius of the circle in the equation

is therefore , making Statement 2 alone sufficient to answer the question - and Statement 1 unhelpful.

### Example Question #24 : Dsq: Calculating The Length Of A Radius

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

Statement 1: Circle 1 has as a diameter ; Circle 2 has as a diameter  is a right triangle.

Statement 2:  and  are the legs of .

EITHER 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.

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

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are true. Since we have no way to determine which, if either, of the legs of  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.

### Example Question #25 : Dsq: Calculating The Length Of A Radius

Right triangle  is inscribed inside a circle. What is the radius of the circle?

Statement 1:

Statement 2:

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.

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.

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.

Explanation:

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   is the hypotenuse, then the radius is half its length; since , the radius is 10.

If  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.