**Video Transcript**

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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!