We are asked what types of problems we can solve using the Pythagorean Theorem. First, let's recall this theorem.

The Pythagorean Theorem
In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

The theorem applies to right triangles, so we can expect to use it in problems that involve the side lengths of right triangles. For example, imagine we need to reach the roof of a building that is $30$ feet tall. We want to know how long a ladder needs to be if we want to put it $10$ feet away from the wall. Let's call the length of the ladder $x .$

The ladder, the wall of the building, and the ground create a right triangle. In this triangle, the legs have lengths of

10

and

30

feet and the hypotenuse has a length

x

feet. We can write an equation using the Pythagorean Theorem.

1 0^{2} + 3 0^{2} = x^{2}

We can solve this equation for

x

to solve our problem.

1 0^{2} + 3 0^{2} = x^{2}

▼

Solve for

x

CalcPow

Calculate power

100 + 900 = x^{2}

AddTerms

Add terms

1000 = x^{2}

SqrtEqn

$LHS = RHS$

1000 = x^{2}

SqrtPowToNumber

$a^{2} = a$

1000 = x

UseCalc

Use a calculator

31.622776 \dots = x

RoundDec

Round to $1$ decimal place(s)

31.6 \approx = x

RearrangeEqn

Rearrange equation

x \approx 31.6

The ladder needs to be at least approximately

31.6

feet long. We can also use the Pythagorean Theorem to solve problems involving rectangles. Since all angles in a rectangle are right angles, two adjacent sides of a rectangle and the diagonal that connects them form a right triangle.

This allows us to use the Pythagorean Theorem for problems involving rectangles or squares. For example, let's consider a park that is being designed. The park is rectangular with sides of $500$ and $800$ feet.

The park will have a path leading from the bottom-left corner of the park to the opposite corner. We want to know how long the path is going to be. Let's call the length of this path $x .$

The park is a rectangle and every angle in a rectangle is a right angle. This means that two adjacent sides of the park and the path form a right triangle.

We can use the Pythagorean Theorem to write an equation for the length of the path

x .

50 0^{2} + 80 0^{2} = x^{2}

Let's solve this equation!

50 0^{2} + 80 0^{2} = x^{2}

▼

Solve for

x

CalcPow

Calculate power

250000 + 640000 = x^{2}

AddTerms

Add terms

890000 = x^{2}

SqrtEqn

$LHS = RHS$

890000 = x^{2}

SqrtPowToNumber

$a^{2} = a$

890000 = x

UseCalc

Use a calculator

943.398113 \dots = x

RoundInt

Round to nearest integer

943 \approx x

RearrangeEqn

Rearrange equation

x \approx 943

The path will be approximately

943

feet long. Finally, we can use the Pythagorean Theorem to solve problems in three dimensions. For example, consider the situation where we want to put a rolled-up poster in a cardboard box.

We want to know how long of a poster can fit in the box. Let's find the length $d$ of the longest poster that would fit in this box.

Notice that we can form a right triangle using the segment with length $d,$ one of the lateral edges of the box, and a diagonal of the base.

We can find the length of $d$ using the Pythagorean Theorem. Before that, we first need to find the length of the diagonal of the base. Let's focus on the base of our box and call the length of the diagonal $x .$

Exercise