Answer to Question #119962 in Geometry for Tina

Question

Develop an activity that explains the concept of pathogen theorem.

Answer 1

1 STEP : We need to make sure the formula

$$\left(a+b\right)^2=a^2+2ab+b^2$$

We will use the area method. Without proof, we believe that we understand the formulas for the area of a rectangle with sides $a$ and $b$ - $S=ab$ , and the area of a square with side $a$ - $S=a^2$

Then,

$$S=\left(a+b\right)^2-\text{large square formula}$$

This square consists of two rectangles and two different squares. Then,

$$\boxed{\left(a+b\right)^2=a^2+2\cdot ab+b^2}$$

Q.E.D.

2 STEP : We pass to the proof of the Pythagorean theorem.

We will also use the area method. We construct a large square with side (a + b) of four rectangular triangles as shown in the figure

$$S=\left(a+b\right)^2=a^2+2ab+b^2-\text{large square formula}$$

This square consists of four rectangular triangles with legs $a$ and $b$ and a square with side $c$ , therefore

$$a^2+2ab+b^2=c^2+4\cdot\frac{1}{2}\cdot a\cdot b\\[0.3cm] a^2+2ab+b^2=c^2+2\cdot a\cdot b\\[0.3cm] \boxed{a^2+b^2=c^2}$$

Q.E.D.