# GMAT Math : Equilateral Triangles

## Example Questions

### Example Question #3 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

Given two equilateral triangles  and , which, if either, has the greater perimeter?

Statement 1:

Statement 2:

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 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.

Explanation:

The area of an equilateral triangle is given by the formula

,

where  is its common sidelength. It follows that the triangle with the greater sidelength has the greater area.

We will let  and  stand for the common sidelength of  and , respectively. The question becomes which, if either, of  and  is the greater.

Statement 1 alone can be rewritten by multiplying:

Therefore, .

Therefore, , the length of one side of  is less than , the length of one side of .

Statement 2 alone can be rewritten as . Again, it follows that .

From either statement alone, it follows that  has the greater sidelength, and, consequently, the greater area.

### Example Question #331 : Geometry

Given two equilateral triangles  and , which, if either, has the greater perimeter?

Statement 1:

Statement 2:  has greater area than .

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 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.

Explanation:

Since an equilateral triangle has three sides of equal measure, the perimeter of an equilateral triangle is three times its sidelength, so the triangle with the greater common sidelength has the greater perimeter.

Statement 1 gives precisely this information; since one side of  is longer than one side of , it follows that  has the longer perimeter.

Statement 2 gives that  has the greater area. Since the area of an equilateral triangle depends only on the common length of its sides, the triangle with the greater area, , must also have the greater sidelength and, consequently, the greater perimeter.

### Example Question #5 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

Give the perimeter of equilateral triangle .

Statement 1:  is a radius of a circle with area .

Statement 2:  is the hypotenuse of a 30-60-90 triangle with area .

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.

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.

Explanation:

Assume Statement 1 alone. To find the radius of a circle with area , use the area formula:

This is also the length of each side of , so its perimeter is three times this, or 24.

Assume Statement 2 alone. If we let  be the length of , then, since this the hypotenuse of a 30-60-90 triangle, by the 30-60-90 Theorem, the legs measure  and . Half the product of their lengths is equal to area , so

.

As before, the sidelength of  is 8 and the perimeter is 24.

### Example Question #6 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

Given two equilateral triangles  and , which has the greater perimeter?

Statement 1:  is the midpoint of .

Statement 2:  is the midpoint of .

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 insufficient 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:

Neither statement alone is enough to determine which triangle has the greater perimeter, as each statement gives information about only one point.

Assume both statements to be true. Since  is the segment that connects the endpoints of two sides of , it is a midsegment of the triangle, whose length is half the length of the side of  to which it is parallel. Therefore, the sidelength of  is half that of , and their perimeters are similarly related. This makes  the triangle with the greater perimeter.

### Example Question #7 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

Which, if either, of equilateral triangles  and , has the greater perimeter?

Statement 1:

Statement 2:

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.

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.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Since the perimeter of an equilateral triangle is three times its common sidelength, comparison of the lengths of the sides is all that is necessary to determine which triangle, has the greater perimeter.

If we let  and  be the common sidelengths of  and , respectively, Statement 1 can be rewritten as the equation . This can be expressed as follows:

Therefore, .

Statement 2 can be rewritten as

Once again,

Since either statement alone establishes that , it follows that  has the longer sides and, consequently, the greater perimeter of the triangles.

### Example Question #8 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

Which, if either, is greater: the perimeter of equilateral triangle  or the circumference of a given circle with center ?

Statement 1: The midpoint of  is inside the circle.

Statement 2: The midpoint of  is on the circle.

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.

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.

Explanation:

For simplicity's sake, we will assume that  has sidelength 1, and, consequently, perimeter 3; these arguments work regardless of the size of the triangle.

We will also need the circumference formula .

Assume Statement 1 alone. Since the midpoint of , which we will call , is inside the circle, the radius of the circle must be greater than  . This makes the circumference at least  times this, or , which is greater than 3.

Assume Statment 2 alone. Since the circle has as a radius the segment from  to the midpoint of the opposite side, it is an altitude of , and the radius is the height of the triangle. By way of the 30-60-90 Theorem, this height is , and the circumference of the circle is  times this, or . This is greater than 3.

Either statement alone establishes that the circumference of the circle is greater than 3, the perimeter of .

### Example Question #2 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

Given three equilateral triangles , and , which has the greatest perimeter?

Statement 1:

Statement 2:

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 insufficient to answer the question.

Explanation:

Assume both statements to be true. From Statement 1, since , it follows that ; since the perimeter of an equilateral triangle is three times the length of one side, it follows that  has perimeter greater than . Similarly, from Statement 2, it follows that  has perimeter greater than . However, there is no way to determine whether  or  has the greater perimeter of the two.

### Example Question #1 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

Given three equilateral triangles , and , which has the greatest perimeter?

Statement 1: A circle with diameter equal to the length of  can be circumscribed about  .

Statement 2: A circle with diameter equal to the length of  can be circumscribed about .

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.

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.

Explanation:

Assume Statement 1 alone, and examine the diagram below, which shows a circle circumscribed about :

The diameter, which is equal to  as given by Statement 1, is greater in length than any chord which is not a diameter - and all sides of  are non-diameter chords. Therefore,  has sides of greater length than , and its perimeter is therefore greater. However, nothing is given about

If Statement 2 alone is assumed, then, similarly,  can be shown to have perimeter greater than that of . But nothing can be determined about .

From the two statements together, however,  has a perimeter greater than those of the other two triangles.

### Example Question #41 : Equilateral Triangles

Which, if either, of equilateral triangles  and , has the greater perimeter?

Statement 1:

Statement 2:

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 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.

Explanation:

If we let  and  be the sidelengths of  and , respectively; the statements can be rewritten as:

Statement 1:

Statement 2:

Since the perimeter of an equilateral triangle is three times its common sidelength, comparison of the lengths of the sides is all that is necessary to determine which triangle, has the greater perimeter. The question can therefore be reduced to asking which of  and , if either, is greater.

Statement 1 alone is not sufficient to yield an answer:

Case 1:

Case 2:

Both cases satisfy Statement 1, but in the first case, , meaning that  has greater sidelength and perimeter than , and in the second case, , meaning the reverse. By a similar argument, Statement 2 is insufficient.

Now assume both statements to be true. The two equations together comprise a system of equations:

Multiply the first equation by 3 and the second by , then add:

Now substitute back:

has the greater sidelength and, consequently, the greater perimeter.

### Example Question #461 : Data Sufficiency Questions

Given an equilateral triangle  and a right triangle  with right angle , which has the greater perimeter?

Statement 1:

Statement 2:

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 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.

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

Explanation:

Assume Statement 1 alone. Since in  is the right angle,  is the hypotenuse, which is longer than either of the two legs -  and . Since, in equilateral triangle , it follows that  and . Consequently

,

and  has the greater perimeter.

Statement 2 alone gives insufficient information, since we know nothing about the second leg or the hypotenuse of . We see this by examining these two cases.

Case 1: Let  be an equilateral triangle of sidelength 6, and  in . If second leg  of  has length 4, then the triangle  is a 3-4-5 triangle, with hypotenuse  having length 5. The perimeter of  is , and the perimeter of  is  has the greater perimeter.

Case 2: Let  be an equilateral triangle of sidelength 6, and  in . If second leg  of  has length 20,  has the greater perimeter on the basis of one side alone.