Surds

UK Secondary (7-11)

Triangle Side Lengths (Investigation)

Lesson

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

- Construction paper
- Protractor with ruler
- Calculator
- Pencil

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$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

- 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.

- 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$0 to $10$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?

- 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.
- What do you notice happened to the length of the hypotenuse?