Pythagorean Theorem
Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of the legs.
Exploration
Often when we state the Pythagorean Theorem, we corrupt our use of language by
saying:
In a right triangle, the square on the hypotenuse is equal to the sum of the
squares on the legs.
1. Investigate this statement by drawing a right triangle, then constructing a square on
the hypotenuse with each side the length of the hypotenuse. Also, construct the
square on each leg that has sides the length of the leg.
(a) What is the relationship between the areas of the three squares?
(b) Explain why the corruption of the language for the Pythagorean Theorem makes
sense.
2. Investigate to find which of the following statements are equivalent to the
Pythagorean Theorem:
In a right triangle, the equilateral triangle on the hypotenuse is equal to the sum of
the equilateral triangles on the other two sides.
In a right triangle, the regular pentagon on the hypotenuse is equal to the sum of the
regular pentagons on the other two sides.
In a right triangle, the regular hexagon on the hypotenuse is equal to the sum of the
hexagons on the other two sides.
(a) Which statements, if any, are equivalent to the Pythagorean Theorem?
(b) What types of figures appear to give statements that are equivalent to the
Pythagorean Theorem?
(c) Write a statement that generalizes this principle.
(d) Explain why (or Prove) each of the statements is equivalent to the Pythagorean
Theorem.
3. Try figures other than regular polygons.
(Try isosceles triangles, rectangles, similar polygons, circles, etc.)
For example:In a right triangle, the semicircle on the hypotenuse is equal to the sum
of the semicircles on the other two sides.
(a) Are there figures other than regular polygons that give statements equivalent to
the Pythagorean Theorem? If yes, what properties do these figures have?
(b) Write a statement that generalizes this principle.
(c) Explain why (or Prove) this statement is equivalent to the Pythagorean Theorem.
Student Activity to Help Students Construct a Regular Polygon Given Two Points that
Determine One Side
1. What is the measure of the base angles of a triangle formed by the center of a regular
pentagon and radii drawn to two consecutive vertices?
2. Write a formula for the measure of a base angle of the triangle formed by the center
of a regular polygon and radii drawn to two consecutive vertices.
3. The measure of the central angle of a circle is 40° and the circle has circumference
equal to 8 centimeters. Find the length of the arc determined by the central angle.
4. Given the measure of a central angle of a circle and the circumference of the circle,
write a formula to find the length of the arc determined by the central angle.
5. Given two points that represent two consecutive vertices of a regular pentagon. Use a
protractor, compass, and straightedge to find the center of a regular pentagon.