## Objectives

- To practice simplifying expressions with radicals
- To practice estimating irrational numbers
- To explore rational and irrational numbers with right triangles

## Materials

- Construction paper
- Protractor with ruler
- Calculator
- Pencil

## Procedure

Work on your own or in pairs.

- On your piece of paper draw the right triangles whose legs have the lengths given. Once you draw the legs use a ruler to connect them and make the hypotenuse.
**Leg 1:** 1.5 units, **Leg 2:** 2.5 units
**Leg 1: **3 units, **Leg 2:** 4 units
**Leg 1: **3.25 units, **Leg 2:** 4.1 units
**Leg 1:** 5.3 units, **Leg 2**: 7.2 units
**Leg 1:** 4.4 units, **Leg 2:** 8.3 units

- For each of the triangles calculate the length of the hypotenuse using the following form of the Pythagorean Theorem:

\text{hypotenuse}^2 = (\text{Leg }1)^2 + (\text{Leg }2) ^2

Remember!

Right triangles are triangles that have a right angle. Use the picture provided to guide your drawings.

## Questions:

- What is the exact length of the hypotenuse for each triangle in simplest form?
- Can any of the hypotenuse lengths be expressed without the radical?
- Convert each of the legs of the triangle to mixed numbers.
- Create a number line going from 0 to 10.
- On your number line indicate where you believe each of the hypotenuses lies.
- Find the decimal representation of each hypotenuse to the nearest hundredth. Use this to check if your placement was right on the number line.
- Are the lengths of the legs of the triangles rational or irrational numbers? Explain.
- Are the lengths of the hypotenuses of the triangles rational or irrational numbers? Explain.
- Based on your two previous answers what do you notice?
- Pick two new leg lengths and find the hypotenuse using the same technique. What do you notice?

## Optional:

- Pick one of the triangles. If you wanted to increase the length of each of the legs by 20% what would be the new lengths of each of the sides? Draw it.
- What do you notice happened to the length of the hypotenuse?