GMAT Math : DSQ: Calculating whether right triangles are similar

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Example Question #146 : Triangles

You are given that $\small \Delta ABC$ and $\small \small \Delta DEF$ are right triangles with their right angles at and , respectively. Is it true that $\small \small \Delta ABC \sim \small \Delta DEF$ ?

1) $\small \small \angle A \cong \angle D$

2) AC = 10 and DF = 20

Possible Answers:

BOTH statements TOGETHER are NOT sufficient to answer the question.

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

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

EITHER Statement 1 or Statement 2 ALONE is 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 $\small \small \small \angle B \cong \angle E$ .

Since Statement 1 tells us that $\small \small \angle A \cong \angle D$ , this sets up the conditions for the Angle-Angle Similarity Postulate, so $\small \small \Delta ABC \sim \small \Delta DEF$ .

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.

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

You are given two right triangles: $\triangle ABC$ with right angle , and $\triangle XYZ$ with right angle .

True or false: $\triangle ABC \sim \triangle XYZ$

Statement 1: AC > BC

Statement 2: XZ < YZ

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.

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:

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 XZ < YZ , $\frac{1}{ XZ}> \frac{1}{YZ}$ .

By the multiplication property of inequality, since

AC > BC and $\frac{1}{ XZ}> \frac{1}{YZ}$ ,

$AC \cdot \frac{1}{ XZ} > BC \cdot \frac{1}{YZ}$

$\frac{AC}{ XZ} > \frac{BC}{YZ}$

Since, by definition, $\triangle ABC \sim \triangle XYZ$ requires that $\frac{AC}{ XZ} = \frac{BC}{YZ}$ , $\triangle ABC \nsim \triangle XYZ$ .

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

You are given two right triangles: $\triangle ABC$ with right angle , and $\triangle XYZ$ with right angle .

True or false: $\triangle ABC \sim \triangle XYZ$

Statement 1: The ratio of the perimeter of $\triangle ABC$ to that of $\triangle XYZ$ is 7 to 6.

Statement 2: $AC \cdot YZ = BC \cdot XZ$

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 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.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 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:

$AC \cdot YZ = BC \cdot XZ$

$\frac{AC \cdot YZ }{BC \cdot YZ}= \frac{BC \cdot XZ}{BC \cdot YZ }$

$\frac{AC }{BC }= \frac{ XZ}{ YZ }$

This establishes that two pairs of corresponding sides are in proportion. Their included angles are both right angles, so $\angle C \cong \angle Z$ , and $\triangle ABC \sim \triangle XYZ$ follows from the Side-Angle-Side Similarity Theorem.

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

You are given two right triangles: $\triangle ABC$ with right angle , and $\triangle XYZ$ with right angle .

True or false: $\triangle ABC \sim \triangle XYZ$

Statement 1: $3 \cdot AB = 4 \cdot XY$

Statement 2: $4 \cdot BC= 3 \cdot YZ$

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:

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:

$3 \cdot AB = 4 \cdot XY$

$\frac{3 \cdot AB}{3 \cdot XY} = \frac{4 \cdot XY}{3 \cdot XY}$

$\frac{ AB}{ XY} = \frac{4 }{3 }$

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

From Statement 2, another ratio can be found:

$4 \cdot BC= 3 \cdot YZ$

$\frac{ 4 \cdot BC}{4 \cdot YZ}=\frac{ 3 \cdot YZ }{4 \cdot YZ}$

$\frac{ BC}{ YZ}=\frac{ 3 }{4 }$

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

$\frac{ AB}{ XY} =\frac{ BC}{ YZ}$

From the two statements together, it can be seen that $\frac{ AB}{ XY} \ne \frac{ BC}{ YZ}$ , so $\triangle ABC \nsim \triangle XYZ$ .

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

You are given two right triangles: $\triangle ABC$ with right angle , and $\triangle XYZ$ with right angle .

True or false: $\triangle ABC \sim \triangle XYZ$

Statement 1: $\angle A$ and $\angle Y$ are complimentary.

Statement 2: $\angle B$ and $\angle X$ are complimentary.

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 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 $\angle A$ and $\angle B$ are a complementary pair, as are $\angle X$ and $\angle Y$ .

If Statement 1 is assumed—that is, if $\angle A$ and $\angle Y$ are a complementary pair—then, since two angles complementary to the same angle—here, $\angle Y$ —must be congruent, $\angle A \cong \angle X$ . Since right angles $\angle C \cong \angle Z$ , $\triangle ABC \sim \triangle XYZ$ 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.

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

You are given two right triangles: $\triangle ABC$ with right angle , and $\triangle XYZ$ with right angle .

True or false: $\triangle ABC \sim \triangle XYZ$

Statement 1: $\angle A \cong \angle X$

Statement 2: $\angle B \cong \angle Y$

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 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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone is true. Then, since $\angle C \cong \angle Z$ , both being right angles, and $\angle A \cong \angle X$ from Statement 1, $\triangle ABC \sim \triangle XYZ$ follows by way of the Angle-Angle Similarity Postulate. A similar argument shows Statement 2 also provides sufficient information.

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

You are given two right triangles: $\triangle ABC$ with right angle , and $\triangle XYZ$ with right angle .

True or false: $\triangle ABC \sim \triangle XYZ$

Statement 1: The ratio of the perimeter of $\triangle ABC$ to that of $\triangle XYZ$ is to .

Statement 2: $\frac{AB}{XY} = \frac{3}{2}$ .

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 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:

AC = 9, BC = 12, AC = 15

XZ = 6, YZ = 8, XY = 10

Case 2:

AC = 9, BC = 12, AC = 15

XZ = 8, YZ = 6, XY = 10

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 $\frac{9+12+15}{6+8+10}= \frac{36}{24} = \frac{3}{2}$ ; and,

$\frac{AC}{XY} = \frac{15}{10}$ .

But in Case 1,

$\frac{AC}{XZ} = \frac{BC}{YZ} = \frac{AB}{XY}$ , since $\frac{9}{6} = \frac{12}{8} = \frac{15}{10} = \frac{3}{2}$ , and the similarity follows by way of the Side-Side-Side Similarity Principle.

In Case 2,

$\frac{AC}{XZ} \ne \frac{AB}{XY}$ , since $\frac{9}{8} \ne \frac{15}{10}$ . This violates the conditions of similarity (note that in both cases, $\triangle ABC \sim \triangle YXZ$ , but this is a different statement).

The two statements together are inconclusive.

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

Given: $\bigtriangleup ABC$ and $\bigtriangleup XYZ$ , where $\angle B$ and $\angle Y$ are right angles.

True or false: $\bigtriangleup ABC \sim \bigtriangleup XYZ$

Statement 1: $m \angle A = 45^{\circ }$

Statement 2: $\bigtriangleup XYZ$ is an isosceles triangle.

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:

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, $m \angle A = 45^{\circ }$ and $\angle B$ is right and measures $90^{\circ }$ .

From Statement 2 alone, $\bigtriangleup XYZ$ is isosceles; the acute angles of an isosceles right triangle must both measure $45^{\circ }$ , so, in particular, $m \angle X = 45^{\circ }$ . Also, it is given that $\angle Y$ is right.

$\angle A \cong \angle X$ and $\angle B \cong \angle Y$ (both of the latter being right angles), and by the Angle-Angle Postulate, $\bigtriangleup ABC \sim \bigtriangleup XYZ$ .

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

Given: Rectangles ABCD and EFGH with diagonals $\overline{AC}$ and $\overline{EG}$ , respectively.

True or false: $\bigtriangleup ABC \sim \bigtriangleup EFG$

Statement 1: $\bigtriangleup ADC \sim \bigtriangleup EHG$

Statement 2: $\angle BAC \cong \angle FEG$

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 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.

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 $\bigtriangleup ABC \cong \bigtriangleup CDA$ and $\bigtriangleup EFG \cong \bigtriangleup GHE$ . Congruent triangles are also similar, so it follows that $\bigtriangleup ABC \sim \bigtriangleup CDA$ and $\bigtriangleup EFG \sim \bigtriangleup GHE$ . Since, by Statement 1, $\bigtriangleup ADC \sim \bigtriangleup EHG$ - or, stated differently, $\bigtriangleup CDA \sim \bigtriangleup GHE$ - by transitivity of similarity,

$\bigtriangleup ABC \sim \bigtriangleup CDA \sim \bigtriangleup GHE \sim \bigtriangleup EFG$ , and

$\bigtriangleup ABC \sim \bigtriangleup EFG$ .

Assume Statement 2 alone. The quadrilaterals are rectangles, so $\angle ABC \cong \angle EFG$ , both being right angles. From Statement 2, $\angle BAC \cong \angle FEG$ , setting up the conditions of the Angle-Angle Postulate; therefore, $\bigtriangleup ABC \sim \bigtriangleup EFG$ .

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

Given: Rectangles ABCD and EFGH with diagonals $\overline{AC}$ and $\overline{EG}$ , respectively.

True or false: $\bigtriangleup ABC \sim \bigtriangleup EFG$

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

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

Possible Answers:

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.

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.

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 $\bigtriangleup ABC \sim \bigtriangleup EFG$

Suppose

AB = CD = 9, AD = BC = 3 .

Then Rectangle ABCD has perimeter

AB + BC + CD + AD = 9 + 3 + 9 +3 = 24

and area

$AB \cdot AD = 9 \cdot 3 = 27$

Now set up two cases with different dimensions for Rectangle EFGH .

Case 1:

EF = GH = 3, EH = FG = 1

The perimeter of Rectangle EFGH is

EF + FG + GH + EH = 3 + 1 + 3 + 1 = 8 ,

one-third that of Rectangle ABCD .

The area of Rectangle EFGH is

$EF \cdot EH = 3 \cdot 1 = 3$ ,

one-ninth that of Rectangle ABCD .

$\frac{AB }{EF} = \frac{9}{3} = 3$

$\frac{BC}{FG} = \frac{3}{1} = 3$

$m \angle B = m \angle F = 90^{\circ }$ ,

so by the Side-Angle-Side Similarity Theorem, $\bigtriangleup ABC \sim \bigtriangleup EFG$ .

Case 2:

EF = GH = 1, EH = FG = 3

The perimeter of Rectangle EFGH is

EF + FG + GH + EH = 1+ 3 + 1 + 3 = 8 ,

one-third that of Rectangle ABCD .

The area of Rectangle EFGH is

$EF \cdot EH = 1 \cdot 3= 3$ ,

one-ninth that of Rectangle ABCD .

$\frac{AB }{EF} = \frac{9}{1} = 1$

$\frac{BC}{FG} = \frac{3}{3} = 1$

Sides are not in proportion, making the statement $\bigtriangleup ABC \sim \bigtriangleup EFG$ false.

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GMAT Math : DSQ: Calculating whether right triangles are similar

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Example Question #146 : Triangles

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