Does the hypotenuse ALWAYS have to be the longest side?
Yes. It is the "c^2" in the equation
c^2 = a^2 + b^2
c has to be greater than both a and b, the right-angle sides.
To understand why the hypotenuse is always the longest side in a right triangle, we can look at the Pythagorean theorem. The Pythagorean theorem states that in any right triangle, the square of the length of the hypotenuse (denoted by c) is equal to the sum of the squares of the lengths of the other two sides (denoted by a and b).
The equation for the Pythagorean theorem is c^2 = a^2 + b^2. This means that the hypotenuse, represented by c, must always be greater than both of the other two sides, represented by a and b.
To verify this, you can compare the squares of the lengths of the sides. Take any right triangle and square the lengths of each side. For example, if Side A is 3 units long and Side B is 4 units long, you can calculate their squares as A^2 = 9 and B^2 = 16.
Now, add these squares together: 9 + 16 = 25. Finally, take the square root of 25, which is 5. This corresponds to the length of the hypotenuse, which is the square root of the sum of the squares of the other two sides (c = √(a^2 + b^2)). In this case, the hypotenuse is 5 units long, which is greater than both Side A (3 units) and Side B (4 units).
So, based on the Pythagorean theorem, the hypotenuse is always the longest side in a right triangle as it must be greater than both of the other two sides.