The Pythagorean Theorem 04:35 minutes

Video Transcript

Transcript The Pythagorean Theorem

Damo, the son of Pythagoras, wants to build a vacation cottage for his mother and father. In keeping with the family business of triangles, Damo decides to incorporate the shape into the design of the cottage. What a perfect time to use his dad's theorem:The Pythagorean Theorem!

Right Triangles and Pythagorean Triples

For the roof of the cottage, Damo will use a triangle design, but what size and shape should the triangle be? Damo's dad spoke of many combinations of integers to create the three sides of right triangles, called Pythagorean triples, but which one is best? Damo thinks back to his dad's class on triangles from school. His dad taught him well! He remembers that right triangles have one and only one right angle, and a right angle is equal to 90°. Even though one angle has to be 90°, there are several different combinations of side lengths that Damo can use.

These special relationships are called Pythagorean Triples. If the integer side lengths of a triangle create a Pythagorean triple, the formula a² + b² = c² will work. How about this triangle, is it a Pythagorean triple? Let's see. The triangle has a right angle, and if you sum the area of the two smaller squares, the sum is equal to the area of the larger square.

Examples for Pythagorean Triples

Damo has a choice of several different roof shapes, but he wants one in the shape of a right triangle, and it must be just right for the cottage. The first Pythagorean triple Damo remembers is 3-4-5. Plugging this into dear ol' dad's theorem gives him, 3² + 4² = 5² ...which is 9 + 16 = 25. This is awesome! No wonder dad gets so excited about right triangles. He knows this one is in the shape of a right triangle, but it's too small.
Next, he tries side lengths 8, 15, and 17. Let’s see, is it a right triangle? Shall we try out the theorem? 8² + 15² = 17². So...64 + 225 = 289. Yup, this one works too, but darn – it looks just like the roof on top of Yorgos’ Yogurt Shop, and that just won’t do for his parents.

There are only a few more roof styles that the roof cutters offer: Damo checks the larger roof, with side measures 7-24-25, first. Does it satisfy the theorem? 7² + 24² = 25². This is the same as 49 + 576 = 625. Yes it works, but it’s just too big.
There's one last option, a triangle with side lengths of 5, 12 and 13. I sure hope this one works 25 + 144 = 169. Yeah! This shape is a right triangle, and it's just the right size and style. So Damo picked a roof that's just right. He also practiced the Pythagorean Theorem and learned some Pythagorean triples. Seems like a win, win to me!

The roof of the cottage looks great, and his dad looks very pleased with the shape. Oh no! Damos forgot that different materials have different densities. Oh Geez. Where was Damo when his dad was teaching the unit on density?? I guess he can still learn something from his dear ol' dad. Join us next time for more triangular adventures with Damo!