GMAT Math : DSQ: Calculating whether right triangles are similar

Study concepts, example questions & explanations for GMAT Math

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Example Question #1 : Dsq: Calculating Whether Right Triangles Are Similar

You are given that  and  are right triangles with their right angles at  and , respectively. Is it true that  ?

1) 

2)  and 

Possible Answers:

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

EITHER Statement 1 or Statement 2 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 NOT sufficient to answer the question.

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

Correct answer:

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

Explanation:

All right angles are congruent, so .

Since Statement 1 tells us that , this sets up the conditions for the Angle-Angle Similarity Postulate, so .

Statement 2 alone only tells us their hypotenuses. Congruence between one pair of angles and the measures of one pair of sides is insufficient information to determine whether two triangles are similar (given one angle, at least two pairs of proportional sides are required).

Therefore, the answer is that Statement 1 alone, but not Statement 2, is sufficient.

Example Question #2 : Dsq: Calculating Whether Right Triangles Are Similar

You are given two right triangles:  with right angle , and  with right angle 

True or false: 

Statement 1: 

Statement 2: 

Possible Answers:

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 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:

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

Explanation:

Each statement alone only gives a relationship between two sides within one triangle, so neither alone answers the question of the similarity of the two triangles.

Assume both statements are true. Then, since ,  

By the multiplication property of inequality, since 

 and ,

Since, by definition,  requires that .

Example Question #3 : Dsq: Calculating Whether Right Triangles Are Similar

You are given two right triangles:  with right angle , and  with right angle 

True or false: 

Statement 1: The ratio of the perimeter of  to that of  is 7 to 6.

Statement 2: 

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:

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

Explanation:

Assume Statement 1 alone. The ratio of the perimeters does not in and of itself establish similarity, since only one angle congruence is known.

Assume Statement 2 alone. The equation can be rewritten as a proportion statement:

This establishes that two pairs of corresponding sides are in proportion. Their included angles are both right angles, so , and  follows from the Side-Angle-Side Similarity Theorem.

Example Question #4 : Dsq: Calculating Whether Right Triangles Are Similar

You are given two right triangles:  with right angle , and  with right angle 

True or false: 

Statement 1: 

Statement 2: 

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.

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.

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. The statement used to find a sidelength ratio:

However, since we only know one sidelength ratio, similarity cannot be proved or disproved.

From Statement 2, another ratio can be found:

Again, since only one sidelength ratio is known, similarty can be neither proved nor disproved.

Assume both statements to be true. Similarity, by definition, requires that 

From the two statements together, it can be seen that , so .

Example Question #5 : Dsq: Calculating Whether Right Triangles Are Similar

You are given two right triangles:  with right angle , and  with right angle 

True or false: 

Statement 1:  and  are complimentary.

Statement 2:  and  are complimentary.

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. 

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The acute angles of a right triangle are complementary, so  and  are a complementary pair, as are  and .

If Statement 1 is assumed—that is, if  and  are a complementary pair—then, since two angles complementary to the same angle—here, —must be congruent, . Since right angles ,   follows by way of the Angle-Angle Similarity Postulate, and Statement 1 turns out to provide sufficient information. By a similar argument, Statement 2 is also sufficient.

Example Question #6 : Dsq: Calculating Whether Right Triangles Are Similar

You are given two right triangles:  with right angle , and  with right angle 

True or false: 

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.

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. 

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone is true. Then, since , both being right angles, and  from Statement 1,  follows by way of the Angle-Angle Similarity Postulate. A similar argument shows Statement 2 also provides sufficient information.

Example Question #7 : Dsq: Calculating Whether Right Triangles Are Similar

You are given two right triangles:  with right angle , and  with right angle 

True or false: 

Statement 1: The ratio of the perimeter of  to that of  is  to .

Statement 2: .

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

Assume both statements are true.

While in two similar triangles, the ratio of the perimeters, given in Statement 1, is indeed equal to that of the ratios of the lengths of the hypotenuses, given in Statement 2, this is not a sufficient condition for similarity. For example:

Case 1: 

Case 2:

In each case, the conditions of the main problem and both statements are met, since:

Both triangles are right - each Pythagorean triple is a multiple of Pythagorean triple 3-4-5;

The ratio of the perimeters is ; and,

.

But in Case 1, 

, since , and the similarity follows by way of the Side-Side-Side Similarity Principle. 

In Case 2, 

, since . This violates the conditions of similarity (note that in both cases, , but this is a different statement).

The two statements together are inconclusive.

Example Question #8 : Dsq: Calculating Whether Right Triangles Are Similar

Given:  and , where  and  are right angles.

True or false: 

Statement 1: 

Statement 2:  is an isosceles triangle.

Possible Answers:

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.

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:

Each of Statement 1 and Statement 2 gives information about only one of the triangles, so neither statement alone is sufficient.

Assume both statements are true. From Statement 1,  and  is right and measures .

From Statement 2 alone,  is isosceles; the acute angles of an isosceles right triangle must both measure , so, in particular,  . Also, it is given that  is right.

 and  (both of the latter being right angles), and by the Angle-Angle Postulate, .

Example Question #9 : Dsq: Calculating Whether Right Triangles Are Similar

Given: Rectangles  and  with diagonals  and , respectively. 

True or false: 

Statement 1: 

Statement 2: 

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:

Refer to the figure below, which gives the two rectangles with their diagonals as described.

Triangles and rectangles

Assume Statement 1 alone. The diagonals of a rectangle bisect the rectangle into congruent triangles, so  and . Congruent triangles are also similar, so it follows that  and . Since, by Statement 1,  - or, stated differently,  - by transitivity of similarity, 

, and

.

 

Assume Statement 2 alone. The quadrilaterals are rectangles, so , both being right angles. From Statement 2, , setting up the conditions of the Angle-Angle Postulate; therefore, .

Example Question #10 : Dsq: Calculating Whether Right Triangles Are Similar

Given: Rectangles  and  with diagonals  and , respectively. 

True or false: 

Statement 1: The perimeter of Rectangle  is three times that of Rectangle .

Statement 2: The area of Rectangle  is nine times that of Rectangle .

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.

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:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

Assume both statements are true. We show the two statements together are insufficent to prove or disprove that 

Suppose 

.

Then Rectangle  has perimeter 

and area 

Now set up two cases with different dimensions for Rectangle .

Case 1:

The perimeter of Rectangle  is 

,

one-third that of Rectangle .

The area of Rectangle  is

one-ninth that of Rectangle .

,

so by the Side-Angle-Side Similarity Theorem, .

 

Case 2:

The perimeter of Rectangle  is 

,

one-third that of Rectangle .

The area of Rectangle  is

one-ninth that of Rectangle .

Sides are not in proportion, making the statement  false.

 

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