DSQ: Calculating if two acute / obtuse triangles are similar - GMAT Quantitative Reasoning

Card 0 of 2

Question

Which of two triangles has greater area, $\Delta ABC$ or $\Delta DEF$ ?

Statement 1: $\angle A \cong \angle D; \angle B \cong \angle E$

Statement 2: $AB = 1.1 \cdot DE$

Answer

Statement 1 alone proves that $\Delta ABC \sim \Delta DEF$ 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 $\Delta ABC$ to the corresponding side in $\Delta DEF$ , so, as the triangles are similar, this ratio is shared by all three pairs of corresponding sides. Since

$\frac{AB }{DE}= 1.1$ ,

1.1 is this common ratio, and the ratio of the area of $\Delta ABC$ to $\Delta DEF$ is $1.1^{2} = 1.21$ , making $\Delta ABC$ the larger triangle.

Compare your answer with the correct one above

Question

True or false:

$\Delta ABC \sim \Delta DE F$

Statement 1: $\frac{AB }{DE}= \frac{BC }{EF}$

Statement 2: $\angle A \cong \angle D$ and $\angle B \cong \angle E$

Answer

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.

Compare your answer with the correct one above

Tap the card to reveal the answer