GRE Quantitative Reasoning - 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?

$4\sqrt{x}$

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:

$b = 4\sqrt{x}$

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

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relaionship cannot be determined from the information given.

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.

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

Gre triangle

The answer cannot be determined from the information given.

$\sqrt{313}$

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.

2 triangles are similar

Triangle 1 has sides 6, 8, 10

Triangle 2 has sides 5 , 3, x

find x

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

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

The two quantities are equal.

Quantity A is greater.

Quantity B is greater.

The relationship cannot be determined from the information given.

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.

How to find the length of the side of a right triangle

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