We are given the following rectangular prism.

The segment with the length of $x$ inches is the diagonal of the rectangular prism. We are asked to find the value of $x .$ Let's start by taking a closer look at the diagram!

The The rectangular prism with the right triangle marked

Notice that the interior diagonal makes a right triangle with the height and the diagonal of the bas eas its legs. If we know these values, we will be able to use the Pythagorean Theorem to find the length of the diagonal. We know that the height of the prism is $15$ inches, but we do not know the length of the diagonal of the base. Let's consider the base of our prism.

We have another right triangle with an unknown hypotenuse. We can use the Pythagorean Theorem to find it.

a^{2} + b^{2} = c^{2}

In the formula,

a

and

b

are the lengths of the legs and

c

is the length of the hypotenuse of a right triangle. We can see that in the

triangle,

the lengths of the legs are

10

inches

and

10

inches .

Let's apply the Pythagorean Theorem to this triangle.

a^{2} + b^{2} = c^{2} ⇕ 10^{2} + 10^{2} = c^{2}

Now we can solve the equation that we got to find the value of

c .

1 0^{2} + 1 0^{2} = c^{2}

CalcPow

Calculate power

100 + 100 = c^{2}

AddTerms

Add terms

200 = c^{2}

SqrtEqn

$LHS = RHS$

200 = c^{2}

CalcRoot

Calculate root

14.142135 \dots = c^{2}

SqrtPowToNumber

$a^{2} = a$

14.142135 \dots = c

RearrangeEqn

Rearrange equation

c = 14.142135 \dots

RoundDec

Round to $1$ decimal place(s)

c \approx 14.1

Since a negative side length does not make sense, we will only consider the positive solution. Therefore, we found that the length of diagonal of the base is about

14.1

inches. Returning to our prism, let's consider the larger right triangle made of the height of our prism, its interior diagonal, and the diagonal of the base.

Let's use the Pythagorean Theorem again to find the length of the diagonal of the prism. We are given a $triangle$ with $a = 15,$ $b \approx 14.1,$ and $c = x .$

Exercise