GMAT Math : Equilateral Triangles

Study concepts, example questions & explanations for GMAT Math

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Example Questions

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

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

Statement 1: 

Statement 2: 

Possible Answers:

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. 

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

Explanation:

Assume Statement 1 alone to be true. Since , it follows that the area of , which is , is greater than that of , which is . However, no information is given about . If Statement alone is assumed, it similarly follows that  has greater area than , but nothing is known about the area of . If both Statements are given, however, then, by transitivity,  has the greatest area of the three.

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

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

Statement 1: 

Statement 2: 

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

Let  and  be the sidelengths of  and , respectively; the statements can be rewritten as:

Statement 1: 

Statement 2: 

Since the area of an equilateral triangle is solely dependent on the length of one side, it follows that the triangle with the greater sidelength has the greater area. The question can therefore be reduced to asking which of  and , if either, is greater.

Assume Statement 1 alone. Then

and  has the greater sidelength. It follows that its area is the greater as well.

Assume Statement 2 alone. Then 

or 

.

However, this is not enough to prove which triangle, if either, has the greater sidelength; if , for example,  or  would make this inequality true. Therefore, it is not clear whether  or , if either, is greater, and it is not clear which triangle has the greater sidelength - and area.

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

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

Statement 1:  is the midpoint of .

Statement 2:  is the midpoint of .

Possible Answers:

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

Explanation:

Neither statement alone is enough to determine which triangle has the greater area, 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 ; it follows that  is the triangle with the greater area.

Example Question #10 : Dsq: Calculating The Area Of An Equilateral Triangle

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

Statement 1: 

Statement 2: 

Possible Answers:

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

Explanation:

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

Statement 1: 

Statement 2: 

Since the area of an equilateral triangle is dependent solely upon the length of each side, comparison of the lengths of the sides is all that is necessary to determine which triangle, has the greater area. 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, as we see by examining these two scenarios:

Case 1: 

Case 2: 

Both cases satisfy Statement 1, but in the first case, , meaning that  has greater sidelength and area 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 second equation by  and add to the first:

            

                 

Now substitute back:

The sidelengths are the same, and, consequently, so are the areas of the triangles.

Example Question #431 : Data Sufficiency Questions

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

Statement 1: 

Statement 2: 

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 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. 

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Since the area of an equilateral triangle is solely dependent on the length of one side, it follows that the triangle with the greater sidelength has the greater area.

Statement 1 alone gives that the length of one side of  is greater than that on one side of .It follows that  has the greater area.

Statement 2 alone gives that , from which it follows that

and 

.

Again, this shows that  has the greater sidelength and the greater area.

Example Question #72 : Triangles

Give the area of equilateral triangle .

Statement 1:  is a diameter of a circle with circumference .

Statement 2:  is a side of a 45-45-90 triangle with area .

Possible Answers:

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.

Correct answer:

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. To find the diameter of a circle with circumference , divide the circumference by  to get . This is also the length of each side of the triangle, so we can get the area using the area formula:

.

Assume Statement 2 alone. A 45-45-90 Triangle has congruent legs, and the area is half the product of their lengths, so if we let  be the common sidelength,

By the 45-45-90 Theorem, the hypotenuse has length  times this, or .

Since it is not given whether  is a leg or the hypotenuse of a right triangle, however, the length of  - and consequently, the area - is not clear.

Example Question #71 : Triangles

Find the area of an equilateral triangle.

  1. A side measures .
  2. An angle measures .
Possible Answers:

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

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

Correct answer:

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

Explanation:

In an equilateral triangle all sides are of the same length and all internal angles measure to 

Statement 1: 

Where  represents the length of the side.

If we're given the side, we can calculate the area:

Statement 2: We don't need the angle to find the area.

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

Consider equilateral triangle .

I) The area of triangle  is .

II) Side  is .

What is the height 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.

Each statement alone is enough to solve the question.

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

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

Correct answer:

Each statement alone is enough to solve the question.

Explanation:

Since is states that we are working with a equilateral triangle we can use the formula for area:

 where  is the side length. Once we have calculated the side length we can then plug that value along with the area into the equation:

 and solve for h.

Consider that equilateral triangles have equal sides. This means we can make ABC into two smaller triangles with hypotenuse of 13 and base of 6.5. We can use that to find the height. We can find the height using statement II.

Therefore, both statements alone are sufficient to solve the question.

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

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What is the length of the height of  ?

(1) ,  is the midpoint of 

(2) 

Possible Answers:

Statement 2 alone is sufficient

Statement 1 alone is sufficient

Statements 1 and 2 taken together are not sufficient

Each statement alone is sufficient

Both statements together are sufficient

Correct answer:

Both statements together are sufficient

Explanation:

Firslty, we would need to have the length of the other sides of the triangles to calculate the height. Information about the angles could also be able to see whether the triangle is a special triangle.

From statement one we can say that triangle ABC is an equilateral triangle, since D is the mid point of the the basis. Moreover knowing , we can see that angle  is 60 degrees. Since D, the basis of the height is the midpoint it follows that  is also 60 degrees. Therefore  is also 60 degrees. Hence the triangle is equilateral. However, we don't know the length of any of the side.

 

Statement 2 gives us the piece of missing information. And alone statement 2 doesn't help us find the height.

It follows that both statements together are sufficient.

Example Question #81 : Triangles

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The equilateral triangle is inscribed in the circle. What is the length of the height?

(1) The center of the circle is at  of the vertices A, B and C.

(2) .

Possible Answers:

Statements 1 and 2 taken together are not sufficient.

Statement 2 alone is sufficient.

Statement 1 alone is sufficient.

Both statements together are sufficient.

Each statement alone is sufficient.

Correct answer:

Statements 1 and 2 taken together are not sufficient.

Explanation:

To be able to answer the question, we would need information about the radius or about the sides of the triangle.

Statement 1 tells us that the center of the circle is at   of the vertice. However this is a property and it will be the same in any equilateral triangle inscribed in a circle, indeed, the heights, whose intersection is the center of gravity, all intersect at  of the vertices.

Statement 2 also tells us something that we could have known from the properties of equilateral triangles. Indeed, equilateral triangles have all their 3 angles equal to .

Even by taking both statements together, we can't tell anything about the lengths of the height. Therefore the statements are insufficient.

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