Activity6.4: Measurements that aren't Rational

Purpose: To discover irrational numbers through geometric figures.

Materials: Graph paper or dot paper Links to an external site., colors, and a ruler, or a geoboard app Links to an external site. and word processor.

Example 1: Make a right triangle with a base of 1 and a height of 3. Find the perimeter and area. Make believe that lengths between points vertically and horizontally are 1 cm.

1 + 3 + how long is the diagonal?

The Pythagorean theorem tells us:

$c^2=1^2+3^2$

$c^2=1+9$

$c^2=10$

$c=\sqrt{10}$

Perimeter = 1 +3 +√10 = $4+\sqrt{10}$ cm,
or about 7.16 cm.

Area = 1x3/2 = $1.5\:cm^2$

Example 2:

Make another right triangle with a base of 1 and a height of 3. Find the perimeter and area.

To find the perimeter of this green triangle we must consider the right triangles that it's diagonals form.

The left edge is part of a right triangle with a base of 2 and a height of 3, so its length is √2^2+3^2 = √13.

The right edge is part of a right triangle with a base of 1 and a height of 3, so its length is √1^2+3^2 = √10.

So the perimeter is (1+ √13 + √10) cm or about 7.77 cm, and the area is (1 cm)x(3 cm)/2 = 3/2 cm².

Your turn:

1. Make 3 different triangles with a base of 2 and a height of 3 on a geoboard or graph paper. One of your triangles could be a right triangle, the other two will not be. Find the perimeter of each triangle and then the area of each.
   
   
2. Make a triangle with a perimeter of 12 cm using the geoboard. This will take several attempts, that is, you will have to experiment.