26) The standard: CCSS.MATH.CONTENT.8.G.B.6

Explain a proof of the Pythagorean Theorem and its converse.

The question:

Timmy is palling around in Denver with Chet and Chad. They drive due South on a street for three miles and then change directions and drive on a new road for 4 miles. Chad says his GPS tells him that they are exactly 5 miles from where they started and somewhat Southeast from where they started. Chad says he can prove mathematically that the direction they traveled the 4 mile road on was due East. Timmy and Chet get in an argument with Chad about it saying, “Pythagoras had some things going for him but wasn’t all that bright.” Chad is getting ganged up on in this argument by Timmy and Chet. Can you help Chad out by writing a simple proof of the Pythagorean Theorem and also writing down what it’s converse says?

The answer:

Students should be able to draw a simple proof of the Pythagorean Theorem using a 3, 4, 5 right triangle and by drawing squares for the area of 3², 4², and 5² (see below). The Converse of the Pythagorean Theorem states that if you have a triangle formed and the sides squared equal the other side squared you have a right triangle. Therefore Chad was right because they would have had to been on a road at an angle of 90 degrees to the first road they were on (and due East is 90 degrees to due South and the only way they could end up somewhat Southeast of where they started).

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