Help with this one!!! Show steps pls :)

Pls help!!!!!!!! Will give brainliest!!!!!!!!!!!

prohojiy [21]

Okay so lets look at the equation given
$a ^{2} + {b}^{2} = {c}^{2}$
this is the Pythagorean theorem, this equation holds true to all right angle triangles.

for the "a" and "b" terms must be the two shortest lengths and "c" must be the longest length.
why? because "c" represents the hypotenuse in this theorem and the hypotenuse is by definition the longest side of a right triangle.

(side note: always draw a picture, it helps a lot if you doodle a bit to get an idea of what you are working with)

we are given three lengths : 3 , 4, and 5 (im ignoring the measurements for now)

we can look at it two ways with a picture and through algebra

try to draw a right triangle with the given info. you'll see there are only two cases were this is possible the picture above is one case where this is not possible

now we can look at this algebraically
$a = 3 \\ b = 4 \\ c = 5$
(but "a" and "b" can swap lengths)

now with the Pythagorean theorem we can see how this plays out
${a}^{2} + {b}^{2} = {c}^{2} \\ {3 }^{2} + {4}^{2} = {5 }^{2} \\ 9 + 16 = 25 \\ 25 = 25$
and heres with "a" and "b" lengths swapped
${4}^{2} + {3}^{2} = {5}^{2} \\ 16 + 9 = 25 \\ 25 = 25$
now here's a case when we let 5 , our longest length , become a leg and let one our shorter lengths take the place of "c"

$a = 5 \\ b = 3 \\ c = 4 \\ {5}^{2} + {3}^{2} = {4}^{2} \\ 25 + 9 = 16 \\ 34 = 16 \:$
obviously 34 does not equal 16

hope this helped you out