UK Secondary (7-11)
Triangle Side Lengths (Investigation)
Lesson

## 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.

1. 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$1.5 units, Leg 2: $2.5$2.5 units
• Leg 1: $3$3 units, Leg 2: $4$4 units
• Leg 1: $3.25$3.25 units, Leg 2: $4.1$4.1 units
• Leg 1: $5.3$5.3 units, Leg 2: $7.2$7.2 units
• Leg 1: $4.4$4.4 units, Leg 2: $8.3$8.3 units
2. For each of the triangles calculate the length of the hypotenuse using the following form of the Pythagorean Theorem:

$\text{hypotenuse}^2=(\text{Leg }1)^2+(\text{Leg }2)^2$hypotenuse2=(Leg 1)2+(Leg 2)2

Remember!

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

## Questions:

1. What is the exact length of the hypotenuse for each triangle in simplest form?
2. Can any of the hypotenuse lengths be expressed without the radical?
3. Convert each of the legs of the triangle to mixed numbers.
4. Create a number line going from $0$0 to $10$10.
5. On your number line indicate where you believe each of the hypotenuses lies.
6. Find the decimal representation of each hypotenuse to the nearest hundredth. Use this to check if your placement was right on the number line.
7. Are the lengths of the legs of the triangles rational or irrational numbers? Explain.
8. Are the lengths of the hypotenuses of the triangles rational or irrational numbers? Explain.
• Pick one of the triangles. If you wanted to increase the length of each of the legs by $20%$20% what would be the new lengths of each of the sides? Draw it.