GMAT Math : Triangles

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Example Questions

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

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

Statement 1: AB + DF = 10

Statement 2: BC - DE = 2

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

Since the area of an equilateral triangle is wholly dependent on its common sidelength, comparison of the lengths of the sides is all that is necessary to determine which triangle, if either, has the greater area.

If we let and $y\:$ be the common sidelengths of $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , respectively, Statements 1 and 2, respectively, can be rewritten as:

Statement 1: x+y = 10

Statement 2: x - y = 2 .

The question is now whether x <y , x> y , or x= y .

Statement 1 alone is insufficient to determine which sidelength is greater; both (1,9) and (9,1) are easily seen to be solutions, with x <y in the first case, and x> y in the second. Consequently, it is possible for either triangle to have the greater sidelength and the greater area.

Statement 2 alone, however, is sufficient. If x - y = 2 , if follows that

x - y + y = 2 + y

and

x = y+2

This means that x > y . $\bigtriangleup ABC$ has the greater sidelength and, consequently, the greater area.

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

Which figure, if either, has the greater area: equilateral triangle $\bigtriangleup ABC$ or a given circle with center ?

Statement 1: The midpoint of $\overline{AB}$ is on the circle.

Statement 2: is outside the circle.

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.

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.

Correct answer:

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

Explanation:

The area formula for an equilateral triangle is $A =\frac{s^{2}\sqrt{3}}{4}$ ; that of a circle is $A = \pi r^{2}$ . Both will be used here.

Assume Statement 1 alone. If we let be the radius of the circle, then, since the points on the circle include the midpoint of $\overline{AB}$ , the distance from center to that midpoint is , and the length of $\overline{AB}$ is . The area of the circle is $\pi r^{2}$ , and that of the triangle is

$\frac{(2r)^{2}\sqrt{3}}{4} = \frac{4r^{2}\sqrt{3}}{4} = r^{2} \sqrt{3}$ .

Since $\pi > \sqrt{3}$ , the circle has the greater area.

Assume Statement 2 alone. For simplicity's sake, assume that the triangle has sidelength 1. Then its area is $\frac{1^{2}\sqrt{3}}{4} = \frac{\sqrt{3}}{4}$ . Since we only know that is the center of the circle and is outside it, it follows that the radius must be less than 1. This means that the area of the circle must be less than

$\pi \cdot 1^{2} = \pi$

Since the area falls in the range $\left ( 0, \pi\right )$ , and $0 < \frac{\sqrt{3}}{4} < \pi$ , we cannot tell whether the circle or the triangle has the greater area.

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

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

Statement 1: GH > AC

Statement 2: BC> DE

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

BOTH statements TOGETHER are insufficient 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 GH > AC , it follows that the area of $\bigtriangleup GHJ$ , which is $\frac{(GH)^{2}\sqrt{3}}{4}$ , is greater than that of $\bigtriangleup ABC$ , which is $\frac{(AC)^{2}\sqrt{3}}{4}$ . However, no information is given about $\bigtriangleup DEF$ . If Statement alone is assumed, it similarly follows that $\bigtriangleup ABC$ has greater area than $\bigtriangleup DEF$ , but nothing is known about the area of $\bigtriangleup GHJ$ . If both Statements are given, however, then, by transitivity, $\bigtriangleup GHJ$ has the greatest area of the three.

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

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

Statement 1: DE + EF < AB + BC

Statement 2: DF < AC + BC

Possible Answers:

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.

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:

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

Statement 1: y+y< x+x

Statement 2: y< x+x

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

Assume Statement 1 alone. Then

y+y< x+x

2y < 2x

y < x ,

and $\bigtriangleup ABC$ has the greater sidelength. It follows that its area is the greater as well.

Assume Statement 2 alone. Then

y< x+x

y <2x .

However, this is not enough to prove which triangle, if either, has the greater sidelength; if y = 10 , for example, x= 12 or x= 8 would make this inequality true. Therefore, it is not clear whether or $y\:$ , if either, is greater, and it is not clear which triangle has the greater sidelength - and area.

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

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

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

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

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:

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 $\overline{AB}$ 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 $\bigtriangleup DEF$ ; it follows that $\bigtriangleup DEF$ is the triangle with the greater area.

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

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

Statement 1: $3 \cdot AB + 2 \cdot DE =45$

Statement 2: $2 \cdot BC+ DF= 27$

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.

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:

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: 3 x + 2y =45

Statement 2: 2x+y= 27

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

Statement 1 alone is not sufficient to yield an answer, as we see by examining these two scenarios:

Case 1: x= 3, y= 18

$3 x + 2y=3 \cdot 3 + 2 \cdot 18= 9 + 36=45$

Case 2: x = 13, y = 3

$3 x + 2y =3 \cdot 13 + 2 \cdot 3= 39 + 6=45$

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

3 x + 2y =45

2x+y= 27

Multiply the second equation by and add to the first:

3 x + 2y =45

$\underline{-4x-2y= -54}$

-x=-9

x = 9

Now substitute back:

2 (9)+y= 27

y+18 = 27

y = 9

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

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Example Question #21 : Equilateral Triangles

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

Statement 1: AB = DE + 1

Statement 2: $6 \cdot EF = 5 \cdot BC$

Possible Answers:

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.

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:

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 $\bigtriangleup ABC$ is greater than that on one side of $\bigtriangleup DEF$ .It follows that $\bigtriangleup ABC$ has the greater area.

Statement 2 alone gives that $6 \cdot EF = 5 \cdot BC$ , from which it follows that

$\frac{ 5 \cdot BC}{5} = \frac{6 \cdot EF}{5}$

and

$BC = \frac{6 }{5}\cdot EF$ .

Again, this shows that $\bigtriangleup ABC$ has the greater sidelength and the greater area.

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Example Question #12 : Dsq: Calculating The Area Of An Equilateral Triangle

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

Statement 1: $\overline{AB}$ is a diameter of a circle with circumference $6 \pi$ .

Statement 2: $\overline{BC}$ is a side of a 45-45-90 triangle with area .

Possible Answers:

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.

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:

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 $6 \pi$ , divide the circumference by $\pi$ to get $6 \pi \div \pi = 6$ . This is also the length of each side of the triangle, so we can get the area using the area formula:

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{6^{2}\sqrt{3}}{4} = \frac{36\sqrt{3}}{4} = 9\sqrt{3}$ .

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,

$\frac{1}{2}s^{2} = 9$

$s^{2} = 18$

$s = \sqrt{18} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}$

By the 45-45-90 Theorem, the hypotenuse has length $\sqrt{2}$ times this, or $3\sqrt{2} \cdot \sqrt{2} = 3 \cdot 2 = 6$ .

Since it is not given whether $\overline{BC}$ is a leg or the hypotenuse of a right triangle, however, the length of $\overline{BC}$ - and consequently, the area - is not clear.

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Example Question #321 : Geometry

Find the area of an equilateral triangle.

A side measures .
An angle measures $60^{\circ}$ .

Possible Answers:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

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.

Each statement alone is sufficient to answer the question.

Statements 1 and 2 are not sufficient, and additional data is needed 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 $60^{\circ}$ .

Statement 1: $area=\frac{\sqrt{3}}{4}a^2$

Where represents the length of the side.

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

$area=\frac{\sqrt{3}}{4}\times5^2\approx 10.83$

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

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

Consider equilateral triangle ABC .

I) The area of triangle ABC is 84.5mm^2 .

II) Side is 13 mm .

What is the height of ABC ?

Possible Answers:

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

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

Both statements taken together are sufficient to solve the question.

Each statement alone is enough to solve the question.

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:

$A=\frac{\sqrt3}{4}a^2$ 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:

$A=\frac{1}{1}b\cdot h$ 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.

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GMAT Math : Triangles

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

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

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

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

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

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

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

Example Question #21 : Equilateral Triangles

Example Question #12 : Dsq: Calculating The Area Of An Equilateral Triangle

Example Question #321 : Geometry

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

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