Two sides of a right triangle are equal to 3m and 4m. Find the length of the third side. (Two cases)If the two given sides are both legs, then the length of the third side is _ m. If the two given sides are a leg and the hypotenuse, then the length of the third side is _ m.

Case 1: The length of the third side (hypotenuse) is 5m.


My Reasoning/Work:
We use the Pythagorean theorem (a^2+b^2=c^2), the two legs in the Pythagorean theorem are "a" and "b" and the third leg/the hypotenuse, is "c". [tex]3^2+4^2=c^2[/tex]
Then, we find the squares of 3 and 4, and then add them up. [tex]9+16=25[/tex]
Now that we know that the square of the third side is 25m, we now find the square root of 25. [tex] \sqrt{25}=5[/tex]
The hypotenuse is 5m.


Case 2: The length of the third side (the leg) is [tex] \sqrt{7} [/tex]m.

My Reasoning/Work:
Again, we'll use the Pythagorean theorem. [tex]a^2+b^2=c^2[/tex]
This time, instead of using the "b", we're using the "a" (leg) and the "c" (hypotenuse). Because the hypotenuse is the longest side in a right triangle, we get that "c" is equal to 4m. [tex]3^2+b^2=4^2[/tex]
And now we bring down the exponents. [tex]9+b^2=16[/tex]
We now subtract 9 from both sides to leave the "b^2" alone. [tex]9-9+b^2=16-9 \\ b^2=7[/tex]
Finally, we find the square root of 7 to get the answer. But if you use a calculator to find this, you'll get an annoying answer of decimals (2.645751311064591). So, to simplify this, the answer would be [tex] \sqrt{7}[/tex].