Question

Which of the following measurements represent a right triangle?

1) 3,4,5
2)5,12,14
3)4,√24,8
4)5,7,√74

please help and show ur work!! there can be more than one answer!! I really need help with this question because I am confused on what to do.. /.\

Answer 1

To check whether a triangle is a right triangle given the dimensions, we can apply Pythagorean's theorem. To recall, we have

$a^{2} + b^{2} = c^{2}$

where c is the longest side (or more known as the hypotenuse) and a and b as the shorter sides of the right triangle. All right triangles must satisfy this theorem.

Now, to make this clearer, let's check the first choice: {3, 4, 5}. Substituting the values, we have

$3^{2} + 4^{2} = 5^{2}$
$9 + 16 = 25$
$25 = 25$

This clearly shows that {3, 4, 5} satisfies the theorem and thus confirming that it's a right triangle.

We can use the same reasoning for the rest as shown below.

2) {5, 12, 14}

$5^{2} + 12^{2} \neq14^{2}$
$25 + 144\neq156$
$169\neq156$

3) {4, √24, 8}

$4^{2} + (\sqrt{24})^{2}\neq8^{2}$
$16 + 24\neq64$
$40\neq64$

4) {5, 7, √74}

$5^{2} + 7^{2} = (\sqrt{74})^{2}$
$25 + 49 = 74$
$74 = 74$

As shown, only 1) and 4) satisfies the theorem. Showing that the two triangles are right triangles.

Answer: 1 and 4