Question

2. Which of the following could be side lengths of a right triangle? Circle all that apply.A. 12, 16, 20B. 4.5, 6, 7.5C. 5, 12, 13D. 6, 12, 14E. 5, 7, 10

Answer 1

When we have a right triangle, the hypotenuse c is always greater than the sides a and b.

Remember that the hypotenuse is the side opposed to the right angle.

Using the Pythagorean Theorem, it is known that for a right triangle of sides a, b, and c with hypotenuse c, the following condition is satisfied:

`a^2+b^2=c^2`

Check each option in order to know if those are the sides of a right triangle:

A) 12, 16, 20

Since the longest side is 20, if those were the sides of a right triangle, 20 would be the hypotenuse.

Check if the condition is satisfied. On the left hand side of the equation, we have:

`12^2+16^2=144+256=400`

On the right hand side of the equation:

`20^2=400`

Since 12^2+16^2=20^2, then those are the lenghts of the sides of a right triangle.

B) 4.5, 6, 7.5

Since the longest side is 7.5, check the condition:

`4.5^2+6^2=20.25+36=56.25`
`7.5^2=56.25`
`\text{Since }4.5^2+6^2=7.5^2,\text{ then those are the sides of a right triangle.}`

C) 5, 12, 13

Since 13 is the longest side:

`5^2+12^2=25+144=169`
`13^2=169`
`\text{Since }5^2+12^2=13^2,\text{ then those are the sides of a right triangle.}`

D) 6, 12, 14

Since 14 is the longest side:

`6^2+12^2=36+144=180`
`14^2=196`
`\text{Since 6}^2+12^2\\e14^2,\text{ then those are NOT the sides of a right triangle.}`

E) 5, 7, 10

Since 10 is the longest side:

`5^2+7^2=25+49=74`
`10^2=100`
`\text{Since 5}^2+7^2\\e10^2,\text{ then those are NOT the sides of a right triangle.}`

Therefore, the options which could be the side lenghts of a right triangle are A, B, and C.