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:
[tex]a^2+b^2=c^2[/tex]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:
[tex]12^2+16^2=144+256=400[/tex]On the right hand side of the equation:
[tex]20^2=400[/tex]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:
[tex]4.5^2+6^2=20.25+36=56.25[/tex][tex]7.5^2=56.25[/tex][tex]\text{Since }4.5^2+6^2=7.5^2,\text{ then those are the sides of a right triangle.}[/tex]C) 5, 12, 13
Since 13 is the longest side:
[tex]5^2+12^2=25+144=169[/tex][tex]13^2=169[/tex][tex]\text{Since }5^2+12^2=13^2,\text{ then those are the sides of a right triangle.}[/tex]D) 6, 12, 14
Since 14 is the longest side:
[tex]6^2+12^2=36+144=180[/tex][tex]14^2=196[/tex][tex]\text{Since 6}^2+12^2\ne14^2,\text{ then those are NOT the sides of a right triangle.}[/tex]E) 5, 7, 10
Since 10 is the longest side:
[tex]5^2+7^2=25+49=74[/tex][tex]10^2=100[/tex][tex]\text{Since 5}^2+7^2\ne10^2,\text{ then those are NOT the sides of a right triangle.}[/tex]Therefore, the options which could be the side lenghts of a right triangle are A, B, and C.
