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

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Question

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

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Answer

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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Question

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

All of the following are true except:

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Answer

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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Question

$\Delta ABC \sim \Delta D EF$

.

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

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Answer

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. , 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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Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

All of the following are true except:

Tap to reveal answer

Answer

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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Question

$\Delta ABC \sim \Delta D EF$

.

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

Tap to reveal answer

Answer

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. , 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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Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

All of the following are true except:

Tap to reveal answer

Answer

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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Question

$\Delta ABC \sim \Delta D EF$

.

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

Tap to reveal answer

Answer

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. , 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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Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

All of the following are true except:

Tap to reveal answer

Answer

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 $\Delta DEF$ .

← Didn't Know|Knew It →

Question

$\Delta ABC \sim \Delta D EF$

.

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

Tap to reveal answer

Answer

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. , 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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Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

All of the following are true except:

Tap to reveal answer

Answer

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 $\Delta DEF$ .

← Didn't Know|Knew It →

Question

$\Delta ABC \sim \Delta D EF$

.

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

Tap to reveal answer

Answer

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. , 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, $m \angle F < m \angle D < m \angle E$ .

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

All of the following are true except:

Tap to reveal answer

Answer

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 $\Delta DEF$ .

← Didn't Know|Knew It →

Question

$\Delta ABC \sim \Delta D EF$

.

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

Tap to reveal answer

Answer

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. , 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, $m \angle F < m \angle D < m \angle E$ .

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

All of the following are true except:

Tap to reveal answer

Answer

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

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

The proportions of corresponding sides of similar triangles must be equal. Therefore, . .

is an equilateral triangle. Points are the midpoints of , respectively. is constructed. All of the following are true except:

. Order the angles of from least to greatest measure.

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

The proportions of corresponding sides of similar triangles must be equal. Therefore, . .

is an equilateral triangle. Points are the midpoints of , respectively. is constructed. All of the following are true except:

. Order the angles of from least to greatest measure.

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

The proportions of corresponding sides of similar triangles must be equal. Therefore, . .

is an equilateral triangle. Points are the midpoints of , respectively. is constructed. All of the following are true except:

. Order the angles of from least to greatest measure.

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

The proportions of corresponding sides of similar triangles must be equal. Therefore, . .

is an equilateral triangle. Points are the midpoints of , respectively. is constructed. All of the following are true except:

. Order the angles of from least to greatest measure.

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

The proportions of corresponding sides of similar triangles must be equal. Therefore, . .

is an equilateral triangle. Points are the midpoints of , respectively. is constructed. All of the following are true except:

. Order the angles of from least to greatest measure.

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

The proportions of corresponding sides of similar triangles must be equal. Therefore, . .

is an equilateral triangle. Points are the midpoints of , respectively. is constructed. All of the following are true except:

. Order the angles of from least to greatest measure.

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

The proportions of corresponding sides of similar triangles must be equal. Therefore, . .

is an equilateral triangle. Points are the midpoints of , respectively. is constructed. All of the following are true except:

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

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

All of the following are true except:

$\Delta ABC \sim \Delta D EF$

.

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

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

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

All of the following are true except:

$\Delta ABC \sim \Delta D EF$

.

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

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

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

All of the following are true except:

$\Delta ABC \sim \Delta D EF$

.

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

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

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

All of the following are true except:

$\Delta ABC \sim \Delta D EF$

.

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

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

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

All of the following are true except:

$\Delta ABC \sim \Delta D EF$

.

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

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

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

All of the following are true except:

$\Delta ABC \sim \Delta D EF$

.

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

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

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

All of the following are true except: