GRE Math : How to find the length of the side of a right triangle

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #3 : Right Triangles

2 triangles are similar

Triangle 1 has sides  6, 8, 10

Triangle 2 has sides 5 , 3, x

find x

Possible Answers:

2

4

12

5

9

Correct answer:

4

Explanation:

Draw the triangles

Triangle 1 is a 6,8,10 right triangle with 10 as the hypotenuse

Triangle 2: 3 is half of 6, 5 is half of 10; x must be half of 8

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Varsity_tutors_problem1

Given the diagram, indicate if Quantity A is larger, Quantity B is larger, if they are equal, or if there is not enough information given to determine the relationship. 

Quantity A: \dpi{100} \small x

Quantity B: 7.5

Possible Answers:

The two quantities are equal.

Quantity B is greater.

The relaionship cannot be determined from the information given.

Quantity A is greater.

Correct answer:

Quantity A is greater.

Explanation:

Since this is a 30-60-90 triangle, we know that the length of the side opposite the 60 degree angle is \dpi{100} \small \sqrt{3} times the side opposite the 30 degree angle. Thus, \dpi{100} \small 5\sqrt{3}, which is about 8.66. This is larger than 7.5. 

Example Question #3 : How To Find The Length Of The Side Of A Right Triangle

A right triangle's perimeter is \dpi{100} \small 3+\sqrt{3}. The other two angles of the triangle are 30 degrees and 60 degrees.

Quantity A: The triangle's hypotenuse length

Quantity B: 2

Possible Answers:

The relationship cannot be determined from the information given.

Quantity B is greater.

The two quantities are equal.

Quantity A is greater.

Correct answer:

The two quantities are equal.

Explanation:

The ratio of the sides of a 30-60-90 triangle is \dpi{100} \small x:x{\sqrt{3}}:2x, with the hypotenuse being \dpi{100} \small 2x. Thus, the perimeter of this triangle would be \dpi{100} \small x+x\sqrt{3}+2x=3x\sqrt{3}. Since the triangle depicted in this problem has a perimeter of \dpi{100} \small 3\sqrt{3}\dpi{100} \small x must equal 1, which would make the hypotenuse equal to 2.

Example Question #4 : How To Find The Length Of The Side Of A Right Triangle

If the shortest side of a right triangle has length  and its hypotenuse has length , what is the length of the remaining side?

 

 

Possible Answers:

Correct answer:

Explanation:

Use the Pythagorean theorem, a^2 + b^2 = c^2  , with \dpi{100} \small a=x-4  and  \dpi{100} \small c=x+4, and solve for \dpi{100} \small b.

(x-4)^2 + b^2 = (x+4)^2

Rearrange to isolate \dpi{100} \small b^{2}:

b^2 = (x+4)^2 - (x-4)^2

b^2 = (x+4)(x+4) - (x-4)(x-4)

Use FOIL to multiply out:

b^2 = (x^2 + 8x + 16) - (x^2 - 8x + 16)

Distribute the minus sign to rewrite without parentheses:

b^2 = x^2 + 8x + 16 - x^2 + 8x - 16

Combine like terms:

b^2 = 16x

Take the square root of both sides:

b = 4\sqrt{x}

Example Question #5 : How To Find The Length Of The Side Of A Right Triangle

Given the following triangle, what is the length of the unknown side?

Gre triangle

Possible Answers:

The answer cannot be determined from the information given.

Correct answer:

The answer cannot be determined from the information given.

Explanation:

At first sight, it's tempting to assume this is a right triangle and to thus use the Pythagorean Theorem to find a length of 5 for the missing side.

However, the triangle was not stated to be a right triangle in the problem statement, and no indication was given in the drawing to indicate that it was a right triangle either, such as a square demarcation in the vertex opposite the side measuring 13.

Thus there is not enough information to give the length of the missing side. When taking standardized math tests, be careful making assumptions about information that is not given.

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