DSQ: Calculating the length of the side of an acute / obtuse triangle - GMAT Quantitative Reasoning

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Question

Is the triangle isosceles?

Statement 1: The triangle has vertices A(1,5), B(4,2), and C(5,6).

Statement 2: $\overline{AB} = \sqrt{18}, \overline{AC} = \sqrt{17}, \overline{BC} = \sqrt{17}$

Answer

For a triangle to be isosceles, two of the sides must be equal. To determine wheter this is true, we must have the three side lengths. Statement 2 gives us those three side lengths. However, Statement 1 also gives us all of the information we need by giving us the three vertices. By using the distance formula, we can easily get the three triangle sides from the vertices. Therefore both statements alone are sufficient.

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Question

Note: Figure NOT drawn to scale.

The above shows a triangle $\Delta MCT$ inscribed inside a rectangle $\textrm{Rect} ; RECT$ . is $\Delta MCT$ isosceles?

Statement 1: is the midpoint of $\overline{RE}$ .

Statement 2: $\angle RTM \cong \angle ECM$

Answer

We show Statement 1 alone is sufficient:

If is the midpoint of $\overline{RE}$ , then . Opposite sides of a rectangle are congruent, so ; all angles of a rectangle, being right angles, are congruent, so $\angle MRT \cong \angle MEC$ . This sets up the conditions for the Side-Angle-Side Theorem, and $\Delta MRT \cong \Delta MEC$ . Consequently, , and $\Delta MCT$ is isosceles.

Now, we show Statement 2 alone is sufficient:

If $\angle RTM$ , and $\angle ECM$ are congruent, then $\angle MTC$ and $\angle MCT$ , being complements of congruent angles, are congruent themselves. By the Isosceles Triangle Theorem, $\Delta MCT$ is isosceles.

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Question

Is $\Delta ABC$ isosceles?

Statement 1: $\Delta ABC \sim \Delta DEF$

Statement 2:

Answer

Statement 1 alone does not tell us anything unless we know the relative lengths of the sides of $\Delta DEF$ ; Statement 2 only gives us information about another triangle.

Suppose we assume both statements. Then by similarity,

$\frac{AB}{DE} = \frac{BC}{EF}$ .

Since , then

$\frac{AB}{DE} \cdot DE = \frac{BC}{EF} \cdot EF$ , or

.

This makes $\Delta ABC$ isosceles.

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Question

Which side of $\Delta ABC$ is the longest?

Statement 1: $\angle B$ is an obtuse angle.

Statement 2: $\angle A$ and $\angle C$ are both acute angles.

Answer

If we only know that two interior angles of a triangle are acute, we cannot deduce the measure of the third, or even if it is obtuse or right; therefore, Statement 2 alone does not help us.

If we know that $\angle B$ is an obtuse angle, however, we can deduce that $\angle A$ and $\angle C$ are both acute angles, since at least two interior angles of a triangle are acute. Therefore, we can deduce that $\angle B$ has the greatest measure, and that its opposite side, $\overline{AC}$ , is the longest.

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Question

Which of the three sides of $\Delta ABC$ is the longest?

Statement 1: $m \angle A =m \angle B + 20^{\circ }$

Statement 2: $m \angle C = m \angle A + 100^{\circ }$

Answer

The longest side of a triangle is opposite the angle of greatest measure.

From Statement 1 alone, we can find two possible scenarios with different answers:

Case 1:

$m \angle B = 20 ^{\circ }, m \angle A = 40 ^{\circ }, m \angle C = 120 ^{\circ }$

Case 2:

$m \angle C = 20 ^{\circ }, m \angle B = 70 ^{\circ }, m \angle A = 90 ^{\circ }$

In both cases, $m \angle A =m \angle B + 20^{\circ }$ , but in Case 1, $\overline{AB}$ is the longest side, and in Case 2, $\overline{BC }$ is the longest side.

From Statement 2 alone, however, we know that $m \angle C > 100^{\circ }$ , so $\angle C$ is obtuse and the other two angles are acute. That makes $\overline{AB}$ the longest side.

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Question

True or false: $\Delta ABC$ is scalene.

Statement 1:

Statement 2:

Answer

By definition, a scalene triangle has three noncongruent sides.

If , then and , and the triangle is scalene.

If , then and , but , so the triangle is not scalene.

The two statements together are insufficient.

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Question

True or false: $\Delta ABC$ is scalene.

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

Statement 2: $\overline{AB} cong \overline{BC}$

Answer

Assume both statements are true.

By definition, a scalene triangle has three noncongruent sides. Sides opposite noncongruent angles of a triangle are noncongruent, so as a consequence of Statement 1, $\overline{AC} cong \overline{BC}$ . Statement 2 alone establishes that $\overline{AB} cong \overline{BC}$ . However, the two statements together do not establish whether or not $\overline{AB} cong \overline{AC}$ , so it is not clear whether $\Delta ABC$ is scalene or isosceles.

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Question

True or false: $\Delta ABC$ is scalene.

Statement 1: $\overline{AC} cong \overline{BC}$

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

Answer

By definition, a scalene triangle has three noncongruent sides.

Statement 1 alone states that two sides are noncongruent, but no information is given about whether or not third side $\overline{AB}$ is congruent to either of the other sides.

Assume Statement 2 alone. In a triangle, sides opposite congruent angles are congruent, so it follows that $\overline{AB} \cong \overline{AC}$ . The triangle cannot be scalene.

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