# GMAT Math : Triangles

## Example Questions

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### Example Question #1 : Triangles

Two sides of a triangle are 6 and 6. What is the height of the triangle?

(1) The third side of the triangle is also 6.

(2) One of the angles of the triangle is .

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

EACH statement ALONE is sufficient.

Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

EACH statement ALONE is sufficient.

Explanation:

Statement (1) informs us that the triangle is an equilateral triangle, with all sides equal to 6. Therefore, the height would divide the triangle into two 30-60-90 triangles, which have side lengths in a ratio of . For this triangle the hypotenuse would be 6 and the base would be 3. The height would therefore be . SUFFICIENT

Statement (2) also lets you deduce that the triangle is an equilateral triangle. Since the triangle is either isoscles or equilateral (at least 2 sides are equal), that means that two angles are equal. Therefore, if one angle is  , the other two also must be  . See above. SUFFICIENT

### Example Question #2 : Triangles

Triangle  has height . What is the length of ?

(1) .

(2) .

Statement 2 alone is sufficient.

Statements 1 and 2 together are not sufficient.

Each statements alone is sufficient.

Statement 1 alone is sufficient.

Both statements together are sufficient.

Statements 1 and 2 together are not sufficient.

Explanation:

Since we don't know what type the triangle is, we would not only need information about the lengths of the side but also about the characteristics of the triangle.

Statement 1 gives us the length of a side. However, we can't do anything, since we don't know the length of DC, which would allow us the know BD with the Pythagorean Theorem.

Statement 2 also only gives us information about one side of the triangle. Alone it doesn't allow us to calculate any other length.

Even taken together these statements are insufficient since, we don't know any pair of lengths to use in the Pythagorean Theorem. Even though the triangle looks like a isosceles triangle, it doesn't mean that it is.

### Example Question #3 : Triangles

is a triangle with height . What is the length ?

(1) The triangle has an area of  and .

(2)  and .

Statements 1 and 2 together are insufficient.

Each statement alone is sufficient.

Statement 2 alone is sufficient.

Both statements together are sufficient.

Statement 1 alone is sufficient.

Both statements together are sufficient.

Explanation:

To find the length DC, we need to know AD and AC or any of those two provided that ABC is isosceles or equilateral.

Statement 1 tells us the area of the triangle with information about a part of side AC. Since we don't know properties of the triangle, these other lengths can vest many values, just AC can be 12, 24 or 48. Therefore we don't have enough information.

Statement 2 gives us information about angles of the triangle. From what we are told we can see that the triangle is isosceles. Indeed, we know that  since BD is the height. Therefore . Now, that we know that the triangle is isosceles, we know that AC must be 12, since D is the midpoint of AC. Therefore DC must be 6.

Hence, both statements taken together are sufficient.

### Example Question #4 : Triangles

Which of two triangles has greater area,  or  ?

Statement 1:

Statement 2:

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.

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.

Explanation:

Statement 1 alone proves that  by the Angle-Angle Postulate, but does not prove anything about the sides, which would be needed to answer this question. Statement 2 alone only gives a relationship between one side of each triangle; without any further information, this is insufficient.

The two statements together, however, present sufficient evidence. Statement 1 proves that the triangles are similar. Statement 2 gives the ratio of one side in  to the corresponding side in , so, as the triangles are similar, this ratio is shared by all three pairs of corresponding sides. Since

,

1.1 is this common ratio, and the ratio of the area of  to  is , making  the larger triangle.

### Example Question #5 : Triangles

True or false:

Statement 1:

Statement 2:  and

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.

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.

Explanation:

Statement 1 alone gives a proportion between two pairs of corresponding sides of the triangles. This is not enough to prove the triangles similar without the third side proportionality (SSS Simiilarity statement) or the congruence of the included angles (SAS Similarity statement).

Statement 2 gives two congruencies between corresponding angles, which by the Angle-Angle statement is enough to prove the triangles similar.

### Example Question #1 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

Find the perimeter of the obtuse .

I) .

II) .

Either statement is sufficient to answer the question.

Both statements are needed to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Explanation:

We are told PGN is obtuse, so it has one angle larger than 90 degrees. However, we don't know what that angle is. To find the perimeter we need all three sides.

I) Relates the two shorter sides.

II) Relates the longest side to one of the short sides.

However, we cannot find any of our side lengths, so we cannot find the perimeter.

### Example Question #2 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

Give the perimeter of .

Statement 1:

Statement 2:

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.

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.

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

Explanation:

The perimeter of  is equal to the sum of the lengths of the sides; that is,

From Statement 1 alone, we get

we can add  to both sides to get

However, without any further information, we cannot determine the actual perimeter.

A similar argument shows that Statement 2 alone gives insufficient information as well.

However, suppose we were to multiply both sides of the equation in Statement 1 by 2, then add both sides of Statement 2:

Divide both sides by 3:

Since

,

we can substitute 29 for  and find :

While we cannot find  or  individually, this is not necessary; in the perimeter formula, we can substitute 29 for  and 8 for :

.

### Example Question #3 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

True or false:  and  have the same perimeter.

Statement 1:  is isosceles and  is scalene.

Statement 2:  and .

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.

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

Explanation:

Statement 1 alone provides insufficient information to answer the question, since it is possible for an isosceles triangle, which has two or three sides of equal length, to have perimeter equal to or not equal to a scalene triangle, which has three sides of different lengths. For example, a triangle with sides of length 10, 10, and 12 has perimeter , the same as a triangle with sides of length 9, 10, and 13, since , but a triangle with sides of length 10, 10, and 13 has perimeter .

Statement 2 alone provides insufficient information to answer the question. Since  and , it follows that the perimeter are equal if and only if ;  we are not told whether this is true or false.

Now assume both statements.  is isosceles, so two of its sides have equal length; however, it cannot hold that  ; if so, then, since  and , it would follow that , which contradicts  being scalene. Therefore, either  or . If , then  is congruent to one other side of , and, consequently, one other side of , contradicting  being scalene. Therefore, ; as stated before, the perimeters are equal if and only if , so the perimeters are not equal.

### Example Question #4 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

True or false: The perimeter of  is greater than 24.

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 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.

Explanation:

Statement 1 alone gives insufficient information. By the Triangle Inequality Theorem, the sum of the lengths of the shortest two sides of a triangle must be greater than the length of the longest. Examine these two scenarios:

Case 1:

This triangle satisfies the triangle inequality, since ; its perimeter is

Case 2:

This triangle satisfies the triangle inequality, since ; its perimeter is .

Therefore, Statement 1 alone does not answer whether the perimeter is less than, equal to, or greater than 24.

Assume Statement 2 alone. Again, ; since, by Statement 2, , by substitution, . The perimeter of  is

, and, since , then

The perimeter of  is greater than 24.

### Example Question #5 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

Given  and Square , which one 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 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.

Explanation:

For the sake of simplicity, we will assume the length of each side of the square is 1; this reasoning works independently of the sidelength. The perimeter of the square is, as a result, 4, and the length of each of the diagonals  and  is  times the length of a side, or simply .

The equivalent question becomes whether the perimeter of the triangle is greater than, equal to, or less than 4. The statements can be rewritten as

Statement 1:  - or equivalently,

Statement 2:

Assume Statement 1 alone. By the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third. Therefore,

and

Since from Statement 1, , t

By the Addition Property of Inequality, we can add  to both sides;

The perimeter of the triangle is greater than 4; equivalently,  has greater perimeter than Square .

Assume Statement 2. By similar reasoning, since one side has length , the perimeter is at greater than twice this, or , which is greater than 4, so   has greater perimeter than Square .

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