GMAT Math : Acute / Obtuse Triangles

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Example Question #4 : Dsq: Calculating The Length Of The Hypotenuse Of An Acute / Obtuse Triangle

3rd side triangle

$\measuredangle A=50, \measuredangle B=60$ . What is the measure of c?

(1) a=12

(2) b=5

Possible Answers:

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Correct answer:

Statements (1) and (2) TOGETHER are NOT sufficient.

Explanation:

Since A=50 , B=60 , A+B+C=180 , therefore, c=70 thus making this an acute triangle. Pythagorean theorem will not apply.

With the information in statement 1, we can't determine the lengths of any other sides. Therefore, Statement 1 alone is not sufficient.

With the information in statement 2, we can't determine the lengths of any other sides. Therefore, Statement 2 alone is not sufficient.

Using the Third Side Rule for triangles, the information in statements 1 and 2 together would allow us to determine the range of values for c. 7<c<17 , but this does not provide a definitive value for c. Therefore, Both statements together are not sufficient.

Therefore - the correct answer is Statements (1) and (2) TOGETHER are NOT sufficient.

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Example Question #41 : Triangles

Note: Figure NOT drawn to scale.

What is the area of the above arrow?

Statement 1: x = 13

Statement 2: y = 8

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

EITHER 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.

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

Correct answer:

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

Explanation:

It can already be ascertained from the figure that y = 8 , since the left portion is a rectangle, so Statement 2 is redundant.

We can already calculate the area of the rectangular portion of the arrow:

$16 \cdot8 = 128$

All this is left is to calculate the area of the triangular portion. If we know Statement 1, we can take half the product of the height, which is 13, and the base, which is 3+8+3 = 14 :

$\frac{1}{2} \cdot 13 \cdot 14 = 91$

Add these numbers to get the area of the arrow: 128 + 91 = 219

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Example Question #2 : Dsq: Calculating The Area Of An Acute / Obtuse Triangle

Two of the vertices of a triangle on the coordinate plane are (0,0), (0,7) . What is its area?

Statement 1: The -coordinate of the third vertex is 8.

Statement 2: The -coordinate of the third vertex is 5.

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

The triangle has as its base a vertical line of length 7, so the height of the triangle would be the perpendicular - which in this case is horizontal - distance from that base. Since the base is part of the -axis, this distance is the absolute value of the -coordinate, which is ony given by Statement 1. Statement 2 is irrelevant.

This is illustrated by this diagram:

Triangle

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Example Question #3 : Dsq: Calculating The Area Of An Acute / Obtuse Triangle

What is the area of a triangle on the coordinate plane with two of its vertices at (0,0),(10,0) ?

Statement 1: The -coordinate of its third vertex is 6.

Statement 2: The -coordinate of its third vertex is 8.

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.

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.

EITHER statement ALONE is 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:

The area of a triangle is half the product of its base and its height.

The triangle has as its base a horizontal line of length 10, so the height of the triangle would be the perpendicular - which in this case is vertical - distance from the third vertex to that base. Since the base is part of the -axis, this height is the absolute value of the -coordinate, which is only given by Statement 2. Statement 1 turns out to be irrelevant.

This is illustrated by this diagram:

Triangle

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Example Question #1 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Is the triangle isosceles?

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

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

Possible Answers:

Statements 1 and 2 TOGETHER are NOT sufficient.

EACH statement ALONE is sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement 1 ALONE is sufficient, but statement 2 is not sufficient.

Statement 2 ALONE is sufficient, but statement 1 is not sufficient.

Correct answer:

EACH statement ALONE is sufficient.

Explanation:

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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Example Question #1 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Rect

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: $\angle RTM \cong \angle ECM$

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 insufficient 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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

We show Statement 1 alone is sufficient:

If is the midpoint of $\overline{RE}$ , then MR = MC . Opposite sides of a rectangle are congruent, so RT = EC ; 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, MT = MC , 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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Example Question #292 : Geometry

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.

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:

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

Explanation:

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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Example Question #293 : Geometry

Is $\Delta ABC$ isosceles?

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

Statement 2: DE = EF

Possible Answers:

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.

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

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 DE = EF , then

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

AB = BC .

This makes $\Delta ABC$ isosceles.

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Example Question #294 : Geometry

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

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

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

Possible Answers:

EITHER 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.

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:

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

Explanation:

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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Example Question #295 : Geometry

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

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

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

Possible Answers:

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.

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 insufficient to answer the question.

Explanation:

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, $\overline{AC} \ncong \overline{BC}$ . Statement 2 alone establishes that $\overline{AB} \ncong \overline{BC}$ . However, the two statements together do not establish whether or not $\overline{AB} \ncong \overline{AC}$ , so it is not clear whether $\Delta ABC$ is scalene or isosceles.

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GMAT Math : Acute / Obtuse Triangles

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #4 : Dsq: Calculating The Length Of The Hypotenuse Of An Acute / Obtuse Triangle

Example Question #41 : Triangles

Example Question #2 : Dsq: Calculating The Area Of An Acute / Obtuse Triangle

Example Question #3 : Dsq: Calculating The Area Of An Acute / Obtuse Triangle

Example Question #1 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Example Question #1 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Example Question #292 : Geometry

Example Question #293 : Geometry

Example Question #294 : Geometry

Example Question #295 : Geometry

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