explain a proof of the Pythagorean Theorem using the image.
(The imagine is a square full of triangles)
To prove the Pythagorean Theorem using the image, we start by considering a square filled with triangles. Let's label the sides of the square as a, and the two smaller triangles inside as b and c.
We can see that the square is divided into four equal triangles. Each of these triangles has a right angle, which means they are all right-angled triangles.
Now, we can compare the areas of these triangles to establish a relationship between their sides. The area of a right-angled triangle is given by the formula: (1/2) * base * height.
Let's start with the larger triangle on the left side of the square. Its base and height are b and a respectively, so its area is (1/2) * b * a.
Next, let's move to the top triangle, which is similar to the previous one. It has base b and height c. Therefore, its area is (1/2) * b * c.
Similarly, we can consider the bottom triangle, which also has base b and height c, giving us an area of (1/2) * b * c.
Lastly, we have the small triangle on the right side of the square. Its base and height are a and c respectively, so its area is (1/2) * a * c.
Now, summing up the areas of all four triangles, we have:
(1/2) * b * a + (1/2) * b * c + (1/2) * b * c + (1/2) * a * c
This simplifies to (1/2) * [2b * a + 2b * c] or simply b * (a + c).
On the other hand, the area of the square is a^2. Since the square is divided into four triangles of equal areas, the sum of their areas is equal to the area of the square.
Thus, we have:
b * (a + c) = a^2
Expanding the left side gives:
b * a + b * c = a^2
Rearranging the equation, we get:
b * a = a^2 - b * c
Now, we can replace b * a in the equation with its equivalent value:
a^2 - b * c = a^2
Subtracting a^2 from both sides, we obtain:
- b * c = 0
Dividing both sides by -1 gives:
b * c = 0
Since the lengths of the sides of a triangle cannot be zero, we conclude that b * c must be greater than zero. Therefore,
a^2 = b * a + b * c
And this is the Pythagorean Theorem, where a represents the length of the hypotenuse in a right-angled triangle, while b and c represent the lengths of the other two sides.