Use the image to answer the question.
A figure shows a square made from four right-angled triangles that all have the same dimensions. Each of the four right angled triangles have a height labeled a, a base labeled b, and a hypotenuse labeled c. Sides a and b are positioned so that the right angle creates the four outer corners of the outer square. Each vertex of the inner square divides each side of the outer square in two unequal parts labeled a and b, where a is the shorter part and b is the longer part. Each side of the inner square, labeled c, becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square. The four right-angled triangles are shaded.
Explain a proof of the Pythagorean Theorem using the image.
(4 points)
To prove the Pythagorean Theorem using the image, we can start by considering one of the right-angled triangles formed at the corners of the outer square.
Let's focus on the triangle in the top left corner, where the shorter leg is labeled a, the longer leg is labeled b, and the hypotenuse is labeled c.
Using the properties of a square, we know that the lengths of the sides of the outer square are equal. Therefore, the longer leg of the triangle, b, is equal to the sum of the lengths of the two unequal parts on each side of the inner square.
We can express this as: b = 2a.
Now, applying the Pythagorean Theorem, we have:
c^2 = a^2 + b^2
Substituting b = 2a, we get:
c^2 = a^2 + (2a)^2
Expanding, we have:
c^2 = a^2 + 4a^2
Simplifying, we have:
c^2 = 5a^2
Now, let's consider the other three triangles formed in a similar manner at the corners of the outer square. All of these triangles have sides a, b, and c, and by substituting b = 2a, we will obtain the same equation for each of them.
Therefore, we can conclude that for all right-angled triangles with sides a, b, and c, where b = 2a, the equation c^2 = 5a^2 holds true.
This is a specific example of the Pythagorean Theorem, where the squares of the lengths of the legs (a and b) equal the square of the length of the hypotenuse (c).
Hence, we have proved the Pythagorean Theorem using the given image.