Homework 18 (due 4/17)

Problem 1.

One leg of a right triangle is 2 cm longer than the other leg, and the hypothenuse is 3 cm longer than the shorter leg. Find all sides of the triangle. (Hint: use Pythagorean Theorem.)

Problem 2.

Estimate the number Pi as follows: consider a circle of radius 1. Its circumference is 2Pi.

• Inscribe a regular hexagon into the circle. Find its perimeter. (Hint: divide the hexagon into 6 equilateral triangles.) Let's denote this perimeter p₁.
• Circumscribe a regular hexagon. Find its perimeter. (Hint: divide the hexagon into 6 equilateral triangles. Use the Pythagorean Theorem to find the length of the sides of these triangles.) Let's denote this perimeter p₂.
• Notice that p₁ < 2Pi < p₂. What inequality do you obtain for Pi?

Remark: similar procedure can be done with regular polygons with more than 6 sides. The more sides, the harder and longer calculations, but the better the estimate. As many groups indicated in their projects, Archimedes used polygons with up to 96 sides.