Kurt wants to informally prove the converse of the Pythagorean Theorem by producing some evidence that supports it.
Recall that the converse of the Pythagorean Theorem states:
If a triangle has side lengths a,b,c, such that LaTeX: a^2+b^2\:=\:c^2a 2 + b 2 = c 2 , then the triangle is a right triangle. Kurt is going to construct several triangles with different side lengths. Select the TWO sets of three side lengths below that Kurt can use as evidence to support the converse of the Pythagorean Theorem.
Group of answer choices
20,48,52
4,5,6
7,24,25
9,12,16
Clearly if the three sides given form a right-angled triangle, then the largest side would be the hypotenuse.
So simply test if the sum of the squares of the two smaller sides equal the square of the largest.
e.g. 7,24,25
IS 7^ + 24^2 = 25^2 ??
49 + 576 = 625 ?
625 = 625 ? YES! So 7,24,25 form a right-angled triangle
test the others the same way
To determine which sets of side lengths can be used as evidence to support the converse of the Pythagorean Theorem, we need to check if the side lengths satisfy the equation a^2 + b^2 = c^2.
Let's examine each set of side lengths:
1) 20, 48, 52:
To determine if these side lengths satisfy the Pythagorean Theorem, we plug them into the equation: 20^2 + 48^2 = 400 + 2304 = 2704. Taking the square root of 2704 gives us 52, which is equal to the third side length. Therefore, this set does satisfy the Pythagorean Theorem.
2) 4, 5, 6:
Plugging these side lengths into the equation, we get 4^2 + 5^2 = 16 + 25 = 41. This means that 41 is not equal to 6^2, which is 36. Therefore, this set does not satisfy the Pythagorean Theorem.
3) 7, 24, 25:
Using the equation, 7^2 + 24^2 = 49 + 576 = 625. The square root of 625 is 25, which matches the third side length. Hence, this set satisfies the Pythagorean Theorem.
4) 9, 12, 16:
Let's substitute these side lengths into the equation: 9^2 + 12^2 = 81 + 144 = 225. The square root of 225 is 15, which is not equal to the third side length, 16. Therefore, this set does not satisfy the Pythagorean Theorem.
Based on our analysis, the sets of side lengths that can be used as evidence to support the converse of the Pythagorean Theorem are: 20, 48, 52 and 7, 24, 25.