# GMAT Math : Right Triangles

## Example Questions

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### Example Question #1 : Calculating Whether Right Triangles Are Similar

Which set of side lengths cannot be the side lengths of a right triangle?

Explanation:

For a triangle to be a right triangle, the sides must obey the Pythagorean Theorem. Let's try our options.

3, 4, 5: You should know this is a right triangle without having to do any calculations because it is one of the special triangles that you should remember. But if you didn't, .

28, 45, 53:

45, 55, 75:  . The sides don't follow the Pythagorean Theorem so this can't be a right triangle. This is our answer. Let's check the remaining two sets of sides as well.

48, 64, 80:  . These are pretty big numbers and this math might take a while. Instead of doing these calculations, we could also see if 48, 64, and 80 look like any of the special triangles we know. Let's divide the three numbers by 16. 48/16 = 3, 64/16 = 4, and 80/16 = 5. Then this is just a type of 3,4,5 triangle, which we know is a right triangle.

84, 35, 91:  . Again, these are big numbers to square. Let's divide the three numbers by their greatest common factor, 7. 84/7 = 12, 35/7 = 5, 91/7 = 13. Then this is a 5, 12, 13 triangle, which is another of our special triangles that we know is a right triangle.

### Example Question #1 : Right Triangles

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

Explanation:

In order for two right triangles to be similar, the ratio of their dimensions must be equal. First we can check the ratio of the height to the base for the given triangle, and then we can check each answer choice for the triangle with the same ratio:

So now we can check the ratio of the height to the base for each answer option, in no particular order, and the one with the same ratio as the given triangle will be a triangle that is similar:

The triangle with a height of  and a base of  has the same ratio as the given triangle, so this one is similar.

### Example Question #1 : Triangles

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

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 : Right Triangles

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

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 #1 : Right Triangles

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

None of the above

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:

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

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

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does 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.

EITHER STATEMENT ALONE provides 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.

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 #2 : Triangles

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

Explanation:

We need to use the Pythagorean theorem:

### Example Question #1 : Calculating The Length Of The Side Of A Right Triangle

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?

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 .

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 and the longer side as . Then we are able to set up an equation:

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

### Example Question #9 : Right Triangles

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

The triangle cannot exist.

The triangle is obtuse and isosceles.

The triangle is obtuse and scalene.

The triangle is acute and scalene.

The triangle is acute and isosceles.

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 #10 : Right Triangles

Using the following right traingle, calculate the value of

(Not drawn to scale.)

Explanation:

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

where

Our equation is then:

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