Geometry’s Most Elegant Theorem Pythagorean Theorem Lesson 9.4

Theorem 69: The square of the measure of the hypotenuse of a right triangle is equal to the sum of the squares of the measures of the legs. (Pythagorean theorem)

Theorem 69…

A B C c b a a 2 +b 2 =c 2 Draw a right triangle with squares on each side. Theorem 69…

Proof of Theorem  ACB is a right angle 2.Draw CD  to AB 3.CD is an altitude 4.a 2 = (c-x)c 1. Given 2. From a point outside a line only one  can be drawn to the line. 3. A segment drawn from a vertex of a triangle  to the opposite side is an altitude. 4. In a right triangle with an altitude drawn to the hypotenuse. (leg) 2 =(adjacent leg)(hypotenuse) x C-x c b a D A BC

5.a 2 = c 2 -cx 6.b 2 = xc 7.a 2 +b 2 = c 2 - cx + cx 8.a 2 + b 2 = c 2 5.Distributive property 6.Same as 4 7.Addition property 8.Algebra Proof of Theorem 69 continued…

Theorem 70: If the square of the measure of one side of a triangle equals the sum of the square of the measure of the other two sides, then the angle opposite the longest side is a right angle. If a 2 + b 2 = c 2, then ∆ABC is a right ∆ and  C is the right angle.

If we increase c while keeping the a and b the same length,  C becomes larger. If c is the length of the longest side of a triangle, a 2 +b 2 >c 2, then the ∆ is acute a 2 +b 2 =c 2, then the ∆ is right a 2 +b 2 <c 2, then the ∆ is obtuse

Find the perimeter of a rhombus with diagonals of 6 and 10. Remember that the diagonals of a rhombus are perpendicular bisectors of each other = x = x 2 34 = x 2 Since all sides of a rhombus are congruent, the perimeter is.

Find the altitude of an isosceles trapezoid whose sides have lengths of 10, 30, 10, and 20. Draw in an altitude. Right ΔADE is congruent to right ΔBCF, and DE = FC = ½ (30 – 20) = 5. FE D C BA In ΔADE, x = 10 2 x = 100 x 2 = 75 x = Altitude = x