Triangles Unit Test
15 of 1515 of 15 Items
Question
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)
One possible proof of the Pythagorean Theorem using the given image is as follows:
1. Consider the outer square with side length a+b. This square can be divided into four congruent right-angled triangles, each with side lengths a, b, and c.
2. Look at one of these right-angled triangles. By the Pythagorean Theorem, we know that the sum of the squares of the two shorter sides (a and b) is equal to the square of the hypotenuse (c). This can be written as a² + b² = c².
3. Since all four triangles are congruent, this relationship applies to all four triangles. Therefore, we have a total of four equations: a² + b² = c², a² + b² = c², a² + b² = c², and a² + b² = c².
4. To find the area of the outer square, we can calculate it as (a+b)², which expands to a² + 2ab + b².
5. However, we can also calculate the area of the outer square by adding together the areas of the four right-angled triangles and the inner square. The area of one triangle is (1/2)ab, so the combined area of the four triangles is 2ab. The area of the inner square is c².
6. Therefore, we can equate the two expressions for the area of the outer square: a² + 2ab + b² = 2ab + c².
7. By simplifying the equation, we get a² + b² = c², which is the Pythagorean Theorem.
Thus, the Pythagorean Theorem can be proven using the given image by considering the relationship between the side lengths and areas of the triangles and squares involved.