In the diagram shown of right triangle BAC, m∠A=90, m∠B= 45 and AC = 8, what is the length of BC?

In a right triangle, the sides are related by the Pythagorean theorem, which states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

However, in this case, we have a special type of right triangle, a 45-45-90 triangle, where the angles are 45 degrees, 45 degrees, and 90 degrees. In this type of triangle, the sides are in a specific ratio. The side opposite the 45 degree angle (which we'll call a) is in the ratio 1:1:√2 with the other side of the 45 degree angle (which we'll call b) and the hypotenuse (which we'll call c).

This means that a = b and c = a*√2 or b*√2.

In this case, we know that AC (the hypotenuse) is 8, so we can use the ratio to find BC (which is either a or b in our ratio).

Since c = a*√2, we can solve for a (or b) by dividing both sides by √2.

So, a = c/√2 = 8/√2 = 4√2.

Therefore, the length of BC is 4√2.