Prove+It!

** Wondering why this is true? ** Well, ask and you shall receive!

Notice this figure.



How can we prove that a 2 + b 2 = c 2 ?

Here is one way...

In Figure 2 above, the whole square, which contains 4 blue triangles and a beige square, has side lengths of (a+b). So the area of the whole square is: A = (a+b)(a+b) = **a 2 + 2ab + b 2 **

The small beige square has side lengths of c. So the area of the small beige square is: A = c x c = **c 2 **

The four small blue triangles have a height of a and a length of b, so the area of each triangle is A = 1/2(ab) And the area of all four triangles is: A = 4(1/2)(ab) = **2ab**

The area of the whole square will be the same as the area of the small plus the area of the four triangles. So, **a 2 + 2ab + b 2 = c 2 + 2ab** <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">and once you subtract 2ab from each side, you are left with <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">**a 2 + b 2 = c 2 **

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">Not convinced? Here's another proof:

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;"> The yellow triangle has length a, height b, and hypotenuse c. To prove that a 2 + b 2 = c 2, we must prove that the red square plus the green square is equal to the blue square.

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">We will use the first proof we did for the left square, so we already know that the area of the large square on the left will be: <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">A= **c 2 + 2ab**.

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">The parts that make up the whole square on the right include a small square with side lengths of a, a medium square with side lengths of b, and 4 small triangles with bases of a and heights of b. So the area of the parts that make up the whole square on the right will be: <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">A = **a 2 + b 2 + 2ab**

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">Both large squares have side lengths of (a+b), so both squares have the same area. Because of this, we can set the two areas equal to each other: <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">a 2 + b 2 + 2ab = c 2 + 2ab <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">Then you subtract 2ab from both sides to get: <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">**a 2 + b 2 = c 2 **

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">**Subtracting 2ab is the same as subtracting out the 4 right triangles, so at the squares you are left with c 2 on the left and a 2 + b 2 on the right. This proves that the area of c 2 is the same as the area of a 2 + b 2, which helps us better understand how Figure 3 explains the Pythagorean Theorem.

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">Want to test an example?

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;"> <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">So in this triangle...

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">a=5 <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">b=12 <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">c=13

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">Does a 2 + b 2 = c 2 ? <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">Let's find out!!

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">5 2 + 12 2 = 13 2 <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">25 + 144 = 169 <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">169 = 169 <span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">It works!!! :)

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">5, 12, and 13 form a Pythagorean Triple!

<span style="color: #137272; font-family: Georgia,serif; font-size: 120%;">Want to go back to home?