Explain a proof of the Pythagorean theorem
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's consider a right triangle with sides A, B, and C, where C is the hypotenuse. We want to prove that A^2 + B^2 = C^2.
Proof:
Step 1: Draw a right triangle with sides A, B, and C.
Step 2: Draw squares on each side of the triangle. Place four copies of the right triangle inside the larger square, as shown in the diagram below:
A
___________________
| | |
| | |
B | A | B |
|_____C____|
C
Step 3: Observe that the larger square's area is equal to the sum of the areas of the four right triangles and the smaller squares combined.
Step 4: The area of each right triangle is (1/2) * A * B, and the area of each smaller square is A^2 and B^2.
Step 5: The area of the larger square is given by (A + B)^2 = A^2 + 2AB + B^2.
Step 6: Equate the area of the larger square to the sum of areas from Step 3:
A^2 + B^2 + A^2 + B^2 + (1/2) * A * B + (1/2) * A * B = A^2 + 2AB + B^2
Step 7: Simplify the equation:
2A^2 + 2B^2 + (1/2) * 2AB = A^2 + 2AB + B^2
Step 8: Cancel the terms from both sides:
2A^2 + 2B^2 = A^2 + B^2
Step 9: Move the terms to one side:
2A^2 + 2B^2 - (A^2 + B^2) = 0
Step 10: Simplify further:
A^2 + B^2 = 0
This proves that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. Therefore, A^2 + B^2 = C^2.
Certainly! The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's consider a right triangle with sides of lengths a, b, and c, where c is the hypotenuse. To prove the theorem, we need to show that a^2 + b^2 = c^2.
There are several different ways to prove the Pythagorean theorem, but one common and straightforward approach is by using similar triangles.
1. Start by drawing a right-angled triangle with sides a, b, and c.
2. Divide the triangle into two smaller triangles by drawing an altitude from the right angle to the hypotenuse. This will create two smaller triangles that are similar to the original triangle.
3. Label the length of the altitude as h.
4. Now, we can apply the concept of similarity. Since the two smaller triangles are similar to the original triangle, the corresponding sides are proportional. This means that the ratio of the length of the altitude to the length of the hypotenuse is equal to the ratio of the length of the altitude to the shorter leg: h/a = h/c.
5. By cross-multiplication, we get h^2 = ac.
6. Similarly, for the second smaller triangle, we can write h^2 = bc.
7. Adding the two equations together, we get 2h^2 = ac + bc.
8. Simplify by factoring out an h from the right-hand side: 2h^2 = (a + b)c.
9. Since h is the altitude drawn from the right angle, it is also the geometric mean of the segments of the hypotenuse cut by the altitude. This means that h^2 = ab.
10. Substitute ab for h^2 in the equation from step 8: 2ab = (a + b)c.
11. Divide both sides of the equation by 2: ab = (a + b)c/2.
12. Now, rearrange the equation: (a^2 + 2ab + b^2) = c(a + b)/2.
13. Since a + b is the length of the hypotenuse c, the equation becomes a^2 + b^2 + c^2 = 2c^2.
14. Finally, divide both sides of the equation by 2: (a^2 + b^2)/2 = c^2/2.
15. Simplify to get the desired result: a^2 + b^2 = c^2.
And there you have it! A proof of the Pythagorean theorem using the concept of similar triangles.
Sure, here's a step-by-step explanation of a proof for the Pythagorean theorem:
Step 1: Start with a right triangle. Label the lengths of the two shorter sides as 'a' and 'b', and the length of the hypotenuse as 'c'.
Step 2: Draw squares on all three sides of the triangle. The squares on side 'a' and side 'b' will have areas equal to a^2 and b^2, respectively.
Step 3: The square on the hypotenuse, side 'c', will have an area equal to c^2.
Step 4: Divide the square on the hypotenuse into two smaller squares. Each smaller square will have an area equal to a^2 and b^2, respectively.
Step 5: Notice that the area of the larger square (c^2) is equal to the sum of the areas of the two smaller squares (a^2 + b^2).
Step 6: This can be written as the equation c^2 = a^2 + b^2, which is known as the Pythagorean theorem.
Step 7: The Pythagorean theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Therefore, the Pythagorean theorem is proven.