GMAT Math : Problem-Solving Questions

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #4 : Triangles

Which of the following right triangles is similar to one with a height  of  and a base  of ?

Possible Answers:

Correct answer:

Explanation:

In order for two right triangles to be similar, their height-to-base ratios must be equal. Given a right triangle with a height  and a base , the ratio . The only answer provided with a ratio of  is .

Example Question #4 : Triangles

Which of the following right triangles is similar to one with a height  of  and a base  of ?

Possible Answers:

None of the above

Correct answer:

Explanation:

In order for two right triangles to be similar, their height-to-base ratios must be equal. Given a right triangle with a height  of  and a base  of ,  the ratio . The only answer provided with a ratio of  is .

Example Question #6 : Right Triangles

Given two right triangles  and , with right angles , what is the measure of ?

Statement 1: 

Statement 2: 

Possible Answers:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Corresponding angles of similar triangles are congruent, so Statement 1 alone establishes that  and ; similarly, Statement 2 alone establishes that  and . However, neither statement alone establishes the actual measure of any of the four acute angles. 

Assume both statements are true. From transitivity, it holds that . Specifically, the two acute angles of , one of which is , are congruent. Since  is right, this makes  and isosceles right triangle, and its acute angles, including , have measure 

Example Question #1 : Triangles

What is the side length of a right triangle with a hypotenuse of  and a side of ?

Possible Answers:

Correct answer:

Explanation:

We need to use the Pythagorean theorem:

 

Example Question #491 : Problem Solving Questions

There is a big square that consists of four identical right triangles and a small square. If the area of the small square is 1, the area of the big square is 5, what is the length of the shortest side of the right triangles?

 

Possible Answers:

\dpi{100} \small 5

\dpi{100} \small 2

\dpi{100} \small 3

\dpi{100} \small 1

\dpi{100} \small 4

Correct answer:

\dpi{100} \small 1

Explanation:

The area of the big square is 5, and area of the small square is 1. Therefore, the area of the four right triangles is \dpi{100} \small 5-1=4.

Since the four triangles are all exactly the same, the area of each of the right triangle is 1. We know that the longer side is 2 times the shorter side, so we can represent the shorter side as \dpi{100} \small x and the longer side as \dpi{100} \small 2x. Then we are able to set up an equation:

\dpi{100} \small \frac{1}{2}\times x\times 2x=1

Therefore, x, the length of the shortest side of the right triangle, is 1.

Example Question #492 : Problem Solving Questions

A triangle has sides of length 9, 12, and 16. Which of the following statements is true?

Possible Answers:

The triangle is obtuse and isosceles.

The triangle cannot exist.

The triangle is acute and scalene.

The triangle is acute and isosceles.

The triangle is obtuse and scalene.

Correct answer:

The triangle is obtuse and scalene.

Explanation:

This triangle can exist by the Triangle Inequality, since the sum of the lengths of its shortest two sides exceeds that of the longest side: 

The sum of the squares of its shortest two sides is less than the square of that of its longest side:

This makes the triangle obtuse.

Its sides are all of different measure, which makes the triangle scalene as well.

Obtuse and scalene is the correct choice.

Example Question #7 : Right Triangles

Using the following right traingle, calculate the value of 

5

(Not drawn to scale.)

Possible Answers:

Correct answer:

Explanation:

We can determine the length of the side by using the Pythagorean Theorem:

where 

Our equation is then: 

Example Question #4 : Calculating The Length Of The Side Of A Right Triangle

Calculate the length of the side of the following right triangle.

6

(Not drawn to scale.)

Possible Answers:

Correct answer:

Explanation:

We can calculate the length of the side by using the pythagorean theorem: 

where our values are 

we can then solve for :

Example Question #12 : Triangles

A right triangle has a hypotenuse of 13 and a height of 5. What is the length of the third side of the triangle?

Possible Answers:

Correct answer:

Explanation:

In order to find the length of the third side, we need to use the Pythagorean theorem. The hypotenuse, c, is 13, and the height, a, is 5, so we can simply plug in these values and solve for b, the length of the base of the right triangle:

Example Question #11 : Triangles

The hypotenuse of a  triangle has length . Which of the following is equal to the length of its shortest leg?

Possible Answers:

Correct answer:

Explanation:

The shortest leg of a  triangle is one half the length of its hypotenuse. In this triangle, it is

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