Drag the points to change the segments on the applet.

Drag one of the sliders. How would you classify the triangle that forms?

What are the angle measures of the triangle?

If the length of the base is 1 unit, what are the lengths of the hypotenuse and the other leg?

Write out the measure of each side length and the opposite angle's measure. What do you notice?

Repeat questions 1-3 with the other triangle.

While the **Pythagorean theorem** can apply to any kind of right triangle, there are particular types of right triangles whose side lengths and angles have helpful properties.

We will look at two **special right triangles**.

The 45 \degree- 45 \degree- 90\degree triangle is an **isosceles right triangle.**

The 30 \degree- 60 \degree- 90\degree triangle can be created by starting with an equilateral triangle and constructing an altitude.

When solving problems involving special right triangles, radicals are often involved. When dividing by a radical, answers should be given with a **rationalized denominator**. A rationalized denominator is a denominator that does not contain a radical.

To rationalize the denominator that contains a radical, we multiply the numerator and denominator by the radical term from the denominator.

\displaystyle \dfrac{6}{\sqrt{2}} | \displaystyle = | \displaystyle \dfrac{6}{\sqrt{2}}\cdot \dfrac{\sqrt{2}}{\sqrt{2}} | Use that \dfrac{\sqrt{2}}{\sqrt{2}}=1 to make an equivalent fraction |

\displaystyle = | \displaystyle \dfrac{6\sqrt{2}}{2} | Use that \sqrt{2}\cdot\sqrt{2}=2 to make the denominator rational | |

\displaystyle = | \displaystyle \dfrac{\cancel{6}\sqrt{2}}{\cancel{2}} | Divide out the common factor of 2 | |

\displaystyle = | \displaystyle 3\sqrt{2} | Write in simplest form |

We can check on the calculator that \dfrac{6}{\sqrt{2}}=3\sqrt{2}.

Consider the triangle below:

a

Find the length of the hypotenuse.

Worked Solution

b

Find and justify the ratio of proportionality between the side lengths of any 45 \degree- 45 \degree- 90\degree triangle.

Worked Solution

Consider the triangle below:

a

Find the height of the triangle.

Worked Solution

b

Find and justify the ratio of proportionality between the side lengths of any 30 \degree- 60 \degree- 90\degree triangle.

Worked Solution

Consider the triangle below:

a

Find the exact value of a.

Worked Solution

b

Find the exact value of c.

Worked Solution

Find the value of \theta.

Worked Solution

Norman is building a backyard playground for this dog and wants to include the King of the Hill obstacle. He wants one side of the ramp to be 6 feet long and for the ramp to make a 45 \degree angle with the ground.

Determine x, the maximum height of the ramp.

Worked Solution

Find the exact value of each variable in the diagram shown.

Worked Solution

Idea summary

We can use special right triangles to find missing side lengths or angles in particular right triangles.

The 45 \degree- 45 \degree- 90\degree triangle can be drawn by creating an **isosceles right triangle ** with leg lengths of 1 or x.

The 30 \degree- 60 \degree- 90\degree triangle can be drawn by creating an equilateral triangle with side lengths of 2 or 2x and then bisecting the top vertex with an altitude and looking only at one half.