GMAT Math : Equilateral Triangles

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Example Questions

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

Given two equilateral triangles $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , which, if either, has the greater perimeter?

Statement 1: $\frac{AC}{DE} = \frac{5}{6}$

Statement 2: AB = DF - 4

Possible Answers:

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.

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.

Explanation:

The area of an equilateral triangle is given by the formula

$a= \frac{s^{2}\sqrt{3}}{4}$ ,

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

We will let and $y\!$ stand for the common sidelength of $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , respectively. The question becomes which, if either, of and $y\:$ is the greater.

Statement 1 alone can be rewritten by multiplying:

$\frac{x}{y} = \frac{5}{6}$

$\frac{x}{y} \cdot y = \frac{5}{6} \cdot y$

$x= \frac{5}{6} \cdot y$

Therefore, x< y .

Therefore, , the length of one side of $\bigtriangleup ABC$ is less than $y\!$ , the length of one side of $\bigtriangleup DEF$ .

Statement 2 alone can be rewritten as x = y -4 . Again, it follows that x< y .

From either statement alone, it follows that x< y . $\bigtriangleup DEF$ has the greater sidelength, and, consequently, the greater area.

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

Given two equilateral triangles $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , which, if either, has the greater perimeter?

Statement 1: AB > DE

Statement 2: $\bigtriangleup ABC$ has greater area than $\bigtriangleup DEF$ .

Possible Answers:

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.

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 insufficient to answer the question.

Correct answer:

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 $\bigtriangleup ABC$ is longer than one side of $\bigtriangleup DEF$ , it follows that $\bigtriangleup ABC$ has the longer perimeter.

Statement 2 gives that $\bigtriangleup ABC$ 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, $\bigtriangleup ABC$ , must also have the greater sidelength and, consequently, the greater perimeter.

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

Give the perimeter of equilateral triangle $\bigtriangleup ABC$ .

Statement 1: $\overline{AB}$ is a radius of a circle with area $64 \pi$ .

Statement 2: $\overline{BC}$ is the hypotenuse of a 30-60-90 triangle with area $8 \sqrt{3}$ .

Possible Answers:

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.

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.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. To find the radius of a circle with area $6 4 \pi$ , use the area formula:

$\pi r ^{2} = A$

$\pi r ^{2} = 6 4 \pi$

$r ^{2} = 6 4$

r = 8

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

Assume Statement 2 alone. If we let be the length of $\overline{BC}$ , then, since this the hypotenuse of a 30-60-90 triangle, by the 30-60-90 Theorem, the legs measure $\frac{1}{2}s$ and $\frac{1}{2}s \sqrt{3}$ . Half the product of their lengths is equal to area $8 \sqrt{3}$ , so

$\frac{1}{2} \cdot \frac{1}{2}s \cdot \frac{1}{2}s \sqrt{3} = 8\sqrt{3}$

$\frac{1}{8}s^{2} \sqrt{3} = 8\sqrt{3}$

$\frac{1}{8}s^{2} = 8$

$s^{2} = 64$

s = 8 .

As before, the sidelength of $\bigtriangleup ABC$ is 8 and the perimeter is 24.

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Example Question #92 : Triangles

Given two equilateral triangles $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , which has the greater perimeter?

Statement 1: is the midpoint of $\overline{DE}$ .

Statement 2: is the midpoint of $\overline{DF}$ .

Possible Answers:

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.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 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 perimeter, as each statement gives information about only one point.

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

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

Which, if either, of equilateral triangles $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , has the greater perimeter?

Statement 1: AB - DF = 1

Statement 2: DE - BC= -1

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If we let and $y\:$ be the common sidelengths of $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , respectively, Statement 1 can be rewritten as the equation x - y = 1 . This can be expressed as follows:

x - y+ y = 1 + y

x = y+1

Therefore, x > y .

Statement 2 can be rewritten as

y - x = -1

y - x + x= -1 + x

y = x-1

Once again, x > y

Since either statement alone establishes that x > y , it follows that $\bigtriangleup ABC$ has the longer sides and, consequently, the greater perimeter of the triangles.

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

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

Statement 1: The midpoint of $\overline{AC}$ is inside the circle.

Statement 2: The midpoint of $\overline{AB}$ is on the circle.

Possible Answers:

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 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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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

We will also need the circumference formula $C = 2 \pi r$ .

Assume Statement 1 alone. Since the midpoint of $\overline{AC}$ , which we will call , is inside the circle, the radius of the circle must be greater than $AM= \frac{1}{2}$ . This makes the circumference at least $2 \pi$ times this, or $2 \pi \cdot \frac{1}{2} = \pi$ , 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 $\bigtriangleup ABC$ , and the radius is the height of the triangle. By way of the 30-60-90 Theorem, this height is $\frac{\sqrt{3}}{2}$ , and the circumference of the circle is $2 \pi$ times this, or $\frac{\sqrt{3}}{2}\cdot 2 \pi = \pi \sqrt{3}$ . This is greater than 3.

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

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Example Question #9 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

Given three equilateral triangles $\bigtriangleup ABC$ , $\bigtriangleup DEF$ , and $\bigtriangleup GHJ$ , which has the greatest perimeter?

Statement 1: GH > AC

Statement 2: FE > BC

Possible Answers:

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements to be true. From Statement 1, since GH > AC , it follows that $3 \cdot GH > 3 \cdot AC$ ; since the perimeter of an equilateral triangle is three times the length of one side, it follows that $\bigtriangleup GHJ$ has perimeter greater than $\bigtriangleup ABC$ . Similarly, from Statement 2, it follows that $\bigtriangleup DEF$ has perimeter greater than $\bigtriangleup ABC$ . However, there is no way to determine whether $\bigtriangleup DEF$ or $\bigtriangleup GHJ$ has the greater perimeter of the two.

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

Given three equilateral triangles $\bigtriangleup ABC$ , $\bigtriangleup DEF$ , and $\bigtriangleup GHJ$ , which has the greatest perimeter?

Statement 1: A circle with diameter equal to the length of $\overline{AB}$ can be circumscribed about $\bigtriangleup GHJ$ .

Statement 2: A circle with diameter equal to the length of $\overline{GJ}$ can be circumscribed about $\bigtriangleup DEF$ .

Possible Answers:

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.

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:

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

Thingy_5

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 $\bigtriangleup GHJ$ are non-diameter chords. Therefore, $\bigtriangleup ABC$ has sides of greater length than $\bigtriangleup GHJ$ , and its perimeter is therefore greater. However, nothing is given about $\bigtriangleup DEF$ .

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

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

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Example Question #101 : Triangles

Which, if either, of equilateral triangles $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , has the greater perimeter?

Statement 1: $2 \cdot AB + 3 \cdot DE =44$

Statement 2: $3 \cdot BC+ 4 \cdot DF= 62$

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.

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.

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 $y\:$ be the sidelengths of $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , respectively; the statements can be rewritten as:

Statement 1: 2x+3y=44

Statement 2: 3x+4y = 62

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 $y\:$ , if either, is greater.

Statement 1 alone is not sufficient to yield an answer:

Case 1: x= 1, y = 14

AB =1, DE = 14

$2x+ 3y= 2 \cdot 1+ 3 \cdot 14= 2+42=44$

Case 2: x =19, y = 2

$2 \cdot AB + 3 \cdot DE= 2 \cdot 19+ 3 \cdot 2= 38+6=44$

Both cases satisfy Statement 1, but in the first case, x < y , meaning that $\bigtriangleup DEF$ has greater sidelength and perimeter than $\bigtriangleup ABC$ , and in the second case, x> y , 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:

2x+3y = 44

3x+4y = 62

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

6x+9y = 132

$\underline{-6x-8y = -124}$

y = 8

Now substitute back:

2x+3 (8)= 44

2x+24= 44

2x = 20

x= 10

$\bigtriangleup ABC$ has the greater sidelength and, consequently, the greater perimeter.

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Example Question #11 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

Given an equilateral triangle $\bigtriangleup ABC$ and a right triangle $\bigtriangleup DEF$ with right angle $\angle E$ , which has the greater perimeter?

Statement 1: AB = DF

Statement 2: $DE = \frac{1}{2} \cdot AB$

Possible Answers:

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.

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. Since in $\bigtriangleup DEF$ , $\angle E$ is the right angle, $\overline{DF}$ is the hypotenuse, which is longer than either of the two legs - DF > DE and DF > EF . Since, in equilateral triangle $\bigtriangleup ABC$ , AB = BC = AC , it follows that AC > DE and BC > EF . Consequently

AB + BC + AC > DF + DE+EF ,

and $\bigtriangleup ABC$ has the greater perimeter.

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

Case 1: Let $\bigtriangleup ABC$ be an equilateral triangle of sidelength 6, and DE = 3 in $\bigtriangleup DEF$ . If second leg $\overline{EF}$ of $\bigtriangleup DEF$ has length 4, then the triangle $\bigtriangleup DEF$ is a 3-4-5 triangle, with hypotenuse $\overline{DF}$ having length 5. The perimeter of $\bigtriangleup ABC$ is $6 \times 3 = 18$ , and the perimeter of $\bigtriangleup DEF$ is 3+4+5 = 12 . $\bigtriangleup ABC$ has the greater perimeter.

Case 2: Let $\bigtriangleup ABC$ be an equilateral triangle of sidelength 6, and DE = 3 in $\bigtriangleup DEF$ . If second leg $\overline{EF}$ of $\bigtriangleup DEF$ has length 20, $\bigtriangleup DEF$ has the greater perimeter on the basis of one side alone.

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GMAT Math : Equilateral Triangles

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

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

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

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

Example Question #92 : Triangles

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

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

Example Question #9 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

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

Example Question #101 : Triangles

Example Question #11 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

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