GMAT Math : Calculating if two acute / obtuse triangles are similar

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Example Question #41 : Acute / Obtuse Triangles

$\Delta ABC$ is an equilateral triangle. Points D,E,F are the midpoints of $\overline{AB},\overline{BC},\overline{AC}$ , respectively. $\Delta DEF$ is constructed.

All of the following are true except:

Possible Answers:

Each side of $\Delta A BC$ is parallel to a side of $\Delta DEF$ .

$\Delta A BC \sim \Delta DEF$

All of the statements in the other four choices are correct.

The perimeter of $\Delta A BC$ is twice that of $\Delta DEF$ .

The area of $\Delta A BC$ is twice that of $\Delta DEF$ .

Correct answer:

The area of $\Delta A BC$ is twice that of $\Delta DEF$ .

Explanation:

The three sides of $\Delta DEF$ are the midsegments of $\Delta A BC$ , so $\Delta DEF$ is similar to $\Delta A BC$ .

By the Triangle Midsegment Theorem, each is parallel to one side of $\Delta A BC$ .

By the same theorem, each has length exactly half of that side, giving $\Delta A BC$ twice the perimeter of $\Delta DEF$ .

But since the sides of $\Delta A BC$ have twice the length of those of $\Delta DEF$ , the area of $\Delta A BC$ , which varies directly as the square of a sidelength, must be four times that of $\Delta DEF$ .

The correct choice is the one that asserts that the area of $\Delta A BC$ is twice that of $\Delta DEF$ .

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Example Question #2 : Calculating If Two Acute / Obtuse Triangles Are Similar

$\Delta ABC \sim \Delta D EF$

AB = 43, BC = 45, AC = 53 .

Order the angles of $\Delta D EF$ from least to greatest measure.

Possible Answers:

$\angle F, \angle D, \angle E$

The angles of $\Delta D EF$ cannot be ordered from the information given.

$\angle E, \angle F, \angle D$

$\angle E, \angle D, \angle F$

$\angle F, \angle E, \angle D$

Correct answer:

$\angle F, \angle D, \angle E$

Explanation:

In a triangle, the angle of greatest measure is opposite the side of greatest measure, and the angle of least measure is opposite the side of least measure. AB < BC < AC , so their opposite angles are ranked in this order - that is, $m \angle C < m \angle A < m \angle B$ .

Corresponding angles of similar triangles are congruent, so, since $\Delta ABC \sim \Delta D EF$ , $m \angle A= m \angle D , m \angle B= m \angle E, m \angle C = m \angle F$ .

Therefore, by substitution, $m \angle F < m \angle D < m \angle E$ .

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Example Question #1 : Calculating If Two Acute / Obtuse Triangles Are Similar

The triangles are similar. What is the value of x?

Possible Answers:

$Not\ enough\ information$

Correct answer:

Explanation:

The proportions of corresponding sides of similar triangles must be equal. Therefore, $\small \frac{8}{12}\ =\ \frac{10}{x}$ . $\small 8x\ =\ 120; x\ =\ 15$ .

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GMAT Math : Calculating if two acute / obtuse triangles are similar

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #41 : Acute / Obtuse Triangles

Example Question #2 : Calculating If Two Acute / Obtuse Triangles Are Similar

Example Question #1 : Calculating If Two Acute / Obtuse Triangles Are Similar

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