### All GMAT Math Resources

## Example Questions

### Example Question #1 : Equilateral Triangles

Is an equilateral triangle?

Statement 1:

Statement 2: , and is equiangular.

**Possible Answers:**

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.

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

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

If , then

.

This makes an equiangular triangle.

If , and is equiangular, then, since corresponding angles of similar triangles are congruent, has the same angle measures, and is itself equiangular.

From either statement, since all equiangular triangles are equilateral, we can draw this conclusion about .

### Example Question #2 : Equilateral Triangles

True or false: is equilateral.

Statement 1: The perimeter of is .

Statement 2: .

**Possible Answers:**

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

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.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

The two statements together provide insufficient information. A triangle with sides , , and is equilateral and has perimeter ; A triangle with sides , , and is not equilateral and has perimeter .

### Example Question #3 : Equilateral Triangles

is the height of . What is the length of ?

(1)

(2)

**Possible Answers:**

Statement 2 alone is sufficient

Each statement alone is sufficient

Statement 1 alone is sufficient

Both statements together are sufficient

Statements 1 and 2 together are not sufficient

**Correct answer:**

Statements 1 and 2 together are not sufficient

To find the answer we should know more about the characteristics of the triangle, i.e. its angles, sides...

Statement 1 alone is obviously insufficient, since we don't know whether the triangle is equilateral, nothing can be said about AB.

Statement 2 is equally as unhelpful as statement 1, since we don't know whether ABC is of a specific type of triangle.

Taken together, these statements allow us to calculate the length of CB, but we can't go further, because we don't know what is AD.

Therefore statements 1 and 2 are not sufficient even taken together.

### Example Question #4 : Equilateral Triangles

ABC is an equilateral triangle inscribed in the circle. What is the length of side AB?

(1) The area of the circle is

(2) The perimeter of triangle ABC is

**Possible Answers:**

Both statements together are sufficient

Each statement alone is sufficient

Statment 2 alone is sufficient

Statement 1 alone is sufficient

Statements 1 and 2 together are not sufficient

**Correct answer:**

Each statement alone is sufficient

To find the length of the side, we would need to know anything about the lengths in the circle or in the triangle.

From statement 1, we can find the radius of the circle, which allows us to calculate the height of the triangle, since the radius is of the height. And finally since the triangle is equilateral, we can also calculate the length of the sides from the height.

Therefore statement 1 is sufficient.

Statement 2 also gives us useful information, indeed the perimeter is simply three times the length of the sides.

Therefore the final answer is each statement alone is sufficient.

### Example Question #5 : Equilateral Triangles

Find the side length of .

I) has perimeter of .

II) is equal to which is .

**Possible Answers:**

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

Either statement is sufficient to answer the question.

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.

**Correct answer:**

Both statements are needed to answer the question.

I) Tells us the perimeter of the triangle.

II) Tells us that FHT is an equilateral triangle.

Taking these statements together we are able to find the side length by dividing the perimeter from statement I, by 3 since all side lengths of an equilateral are the same by statement II.

### Example Question #6 : Equilateral Triangles

Given equilateral triangle and right triangle , which, if either, is longer, or ?

Statement 1:

Statement 2: is a right angle.

**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.

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.

**Correct answer:**

Assume Statement 1 alone. Since all three sides of are congruent - specifically, - and , it follows by transitivity that . However, no information is given as to whether has length greater then, equal to, or less than , so which of and , if either, is the longer cannot be answered.

Assume Statement 2 alone. Since is the right angle of , is the hypotenuse and this the longest side, so and . However, no comparisons with the sides of can be made.

Now assume both statements are true. as a consequence of Statement 1, and as a consequence of Statement 2, so .

### Example Question #7 : Equilateral Triangles

What is the length of side of equilateral triangle ?

Statement 1: is a diagonal of Rectangle with area 30.

Statement 2: is a diagonal of Square with area 36.

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

An equilateral triangle has three sides of equal measure, so if the length of any one of the three sides can be determined, the lengths of all three can be as well.

Assume Statement 1 alone. is a diagonal of a rectangle of area 30. However, neither the length nor the width can be determined, so the length of this segment cannot be determined with certainty.

Assume Statement 2 alone. A square with area 36 has sidelength the square root of this, or 6; its diagonal, which is , has length times this, or . This is also the length of .

### Example Question #8 : Equilateral Triangles

What is the length of side of equilateral triangle ?

Statement 1: , , and are all located on a circle with area .

Statement 2: The midpoints of all three sides are located on a circle with circumference .

**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.

We demonstrate that either statement alone yields sufficient information by noting that the circle that includes all three vertices of a triangle - described in Statement 1 - is its circumscribed circle, and that the circle that includes all three midpoints of the sides of an equilateral triangle - described in Statement 2 - is its inscribed circle. We examine this figure below, which shows the triangle, both circles, and the three altitudes:

The three altitudes intersect at , which divides each altitude into two segments whose lengths have ratio 2:1. is the center of both the circumscribed circle, whose radius is , and the inscribed circle, whose radius is .

Therefore, from Statement 1 alone and the area formula for a circle, we can find from the area of the circumscribed circle:

From Statement 2 alone and the circumference formula for a cicle, we can find from the circumference of the inscribed circle:

By symmetry, is a 30-60-90 triangle, and either way, , and .

### Example Question #9 : Equilateral Triangles

Given two equilateral triangles and , which, if either, is greater, or ?

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:**

EITHER statement ALONE is sufficient to answer the question.

An equilateral triangle has three sides of equal length, so and .

Assume Statement 1 alone. Since , then, by substitution, .

Assume Statement 2 alone. Since , it follows that , and again by substitution, .

### Example Question #10 : Equilateral Triangles

You are given two equilateral triangles and .

Which, if either, is greater, or ?

Statement 1: The perimeters of and are equal.

Statement 2: The areas of and are equal.

**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, and let be the common perimeter of the triangles. Since an equilateral triangle has three sides of equal length, and , so .

Assume Statement 2 alone, and let be the common area of the triangles. Using the area formula for an equilateral triangle, we can note that:

and ,

so

.

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