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You can use the areas of triangles, squares, and rectangles, or you can use the properties of similar triangles.
See solution.
We can prove the Pythagorean Theorem in two different ways – by using areas of triangles, squares, and rectangles, or by using similarity properties.
We begin by considering a right triangle.
Next, we will consider three more right triangles with the same dimensions and we will arrange them in such a way that we get a large square formed by the legs of the triangles.
The area of the small square is c^2 and the area of each triangle is 12 a b. A = 4* 1/2 a b + c^2 ⇕ A = 2 a b + c^2 On the other side, we can use the lengths of the legs of the triangles to divide the large square into two squares and two rectangles as shown below.
A= a^2 + b^2 + 2 a b
LHS-2 a b=RHS-2 a b
As before, we begin by considering a right triangle.
Next, we will consider the altitude to the hypotenuse.
By applying the Angle-Angle (AA) Similarity Theorem, we get that â–ł ABC ~ â–ł ACD and â–ł ABC ~ â–ł CBD. Then, we can write the following relations.
â–ł ABC ~ â–ł ACD | â–ł ABC ~ â–ł CBD |
---|---|
c/b=b/c- d=a/h | c/a=b/h=a/d |
Multiply
c d= a^2
LHS+ a^2=RHS+ a^2
Rearrange equation