An acute angle is an angle larger than 0 \degree and smaller than 90 \degree
A right angle is an angle measuring exactly 90 \degree
An obtuse angle is an angle larger than 90 \degree and smaller than 180 \degree

A right triangle is a triangle containing an interior right angle.

The Pythagorean theorem and its converse describe how the side lengths of right triangles are related.

Pythagorean theorem

If a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of its legs lengths.

For the given triangle, a^2+b^2=c^2

Right triangle uppercase A uppercase C uppercase B with right angle uppercase C. Uppercase A uppercase B has a length of lowercase c, uppercase A uppercase C has a length of lowercase b, and uppercase B uppercase C has a length of lowercase a.

\displaystyle a^2+b^2=c^2

\bm{a}

is the length of one of the legs (shorter sides) of the right triangle

\bm{b}

is the length of the other leg of the right triangle

\bm{c}

is the length of the hypotenuse of the right triangle

Converse of Pythagorean theorem

If the lengths a, b and c of the three sides of a triangle satisfy the relationship a^2+b^2=c^2, then the triangle is a right triangle.

Pythagorean triple

A set of three non-zero whole numbers a, b, and c such that a^2+b^2=c^2

Example:

3, 4, and 5

Worked examples

Example 1

Find the value of c in the triangle below.

A right triangle with legs of length 14 and 48, and a hypotenuse of length c.

Approach

The triangle is a right triangle so the hypotenuse, c, can be found using the Pythagorean theorem.

Solution

\displaystyle a^2+b^2	\displaystyle =	\displaystyle c^2	Pythagorean theorem
\displaystyle 14^2+48^2	\displaystyle =	\displaystyle c^2	Substitution
\displaystyle 196+2304	\displaystyle =	\displaystyle c^2	Simplify
\displaystyle 2500	\displaystyle =	\displaystyle c^2	Combine like terms
\displaystyle 50	\displaystyle =	\displaystyle c	Square root both sides
\displaystyle c	\displaystyle =	\displaystyle 50	Symmetric property of equality

We have found that c=50.

Reflection

We can use the Pythagorean theorem to solve for missing leg lengths as well.

Notice that in this example, 2500 is a perfect square so our answer for c is an integer. This means \left\{ 14, 48, 50\right\} is a Pythagorean triple.

Example 2

Use the Pythagorean theorem to determine whether the triangle below is a right triangle.

A triangle with side lengths of 9, 16, and 18.

Approach

We want to use the converse of the Pythagorean theorem to determine if the triangle is a right triangle.

Let a and b represent the two shorter side lengths. The hypotenuse will be c.

Once we have labeled the sides, we want to find the value of a^2+b^2 and the value of c^2.

If a^2+b^2=c^2 then the triangle is a right triangle by the converse of the Pythagorean theorem.

If a^2+b^2 \neq c^2 then the triangle is not a right triangle for the same reason.

Solution

Let a=9, b=16, and c=18. Now we can calculate the following:

\displaystyle a^2+b^2	\displaystyle =	\displaystyle 9^2+16^2
\displaystyle {}	\displaystyle =	\displaystyle 337

And

\displaystyle c^2	\displaystyle =	\displaystyle 18^2
\displaystyle {}	\displaystyle =	\displaystyle 324

This is not a right triangle because 337 \neq 324 so a^2+b^2 \neq c^2

Example 3

The size of a computer monitor is often given as the length of its diagonal. Rowena has just bought a 19 inch monitor, meaning that the display has a diagonal length of 19 inches. The screen has a 4:3 width-to-height ratio (aspect ratio).

Determine the width and height of Rowena's new monitor.

Approach

We know the length of the diagonal of the monitor is 19 inches. We also know that the lengths of the width and height are in a ratio of 4:3, but we don't know the exact length of either the width or height.

We can use this information by letting the width be 4x inches and the height be 3x inches, and draw a diagram as follows:

A diagram of a rectangular computer monitor with a width of 4 x, height of 3 x, and diagonal length of 19 inches.

Solution

From the diagram we have drawn, we can use the Pythagorean theorem to find x, and then determine the dimensions of the monitor from there:

\displaystyle \left(4x\right)^2 + \left(3x\right)^2	\displaystyle =	\displaystyle 19^2	Apply Pythagorean theorem
\displaystyle 16x^2 + 9x^2	\displaystyle =	\displaystyle 361	Evaluate powers
\displaystyle 25x^2	\displaystyle =	\displaystyle 361	Combine like terms
\displaystyle x^2	\displaystyle =	\displaystyle \frac{361}{25}	Divide both sides by 25
\displaystyle x	\displaystyle =	\displaystyle \frac{19}{5}	Take the square root of both sides

So we have that x = \dfrac{19}{5} = 3.8 inches. We can now use this to determine the width and height of the monitor:

\text{Width}\, = 4x = 4\left(3.8\right) = 15.2\, \text{in}
\text{Height}\, = 3x = 3\left(3.8\right) = 11.4\, \text{in}

Outcomes

MA.912.GR.1.3

Prove relationships and theorems about triangles. Solve mathematical and real-world problems involving postulates, relationships and theorems of triangles.

MA.912.LT.4.8

Construct proofs, including proofs by contradiction.

MA.912.LT.4.10

Judge the validity of arguments and give counterexamples to disprove statements.

MA.912.T.1.2

Solve mathematical and real-world problems involving right triangles using trigonometric ratios and the Pythagorean Theorem.

8.01 Right triangles and the Pythagorean theorem

Concept summary

Worked examples

Example 1

Approach

Solution

Reflection

Example 2

Approach

Solution

Example 3

Approach

Solution

Outcomes

MA.912.GR.1.3

MA.912.LT.4.8

MA.912.LT.4.10

MA.912.T.1.2

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