Unit 3, Lesson 14
Proving the Pythagorean Theorem
Geometry
14.1
Warm-up
What do you notice? What do you wonder?
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What do you notice? What do you wonder?
A diagram of two congruent squares. The left diagram is composed of 4 congruent right triangles and two smaller squares of different sizes. In the top left corner, two right triangles creating a small rectangle. The small leg is labeled A on the left side of the square. In the top right corner, a small green square. In the bottom right corner, two triangles creating a small rectangle. The small leg is labeled A on the bottom of the square. In the bottom left corner, a larger yellow square, both sides labeled B. The right diagram is composed of the same 4 congruent right triangles and one smaller square inside, on an angle. The long leg of the triangle is labeled b, the short leg of the triangle is labeled a. No labels are provided for the blue square inside.
When Pythagoras proved his theorem, he used the two images shown here. Can you figure out how he used these diagrams to prove that in a right triangle with a hypotenuse of length ?
In any right triangle with legs and and hypotenuse , we know that . We call this the Pythagorean Theorem. But why does it work?
We can use an altitude drawn to the hypotenuse of a right triangle to prove the Pythagorean Theorem.
Right triangle labeled A B C. Right angle points up. Left side labeled A, right side labeled b, arrow on bottom side labeled c. Altitude drawn from right angle to bottom side, labeled f, creating 2 segments on bottom side, left side labeled d creating triangle F D A, right side labeled e, creating triangle F E B.
We can use the Angle-Angle Triangle Similarity Theorem to show that all 3 triangles are similar. Because the triangles are similar, the corresponding side lengths are in the same proportion.
Because the largest triangle is similar to the smaller triangle, . Because the largest triangle is similar to the middle triangle, . We can rewrite these equations as and .
We can add the 2 equations to get that , or . From the original diagram we can see that , so , or .