Question

Let ABCD be a square, and let M and N be the midpoints of BC and CD, respectively. The value of \sin \angle MAN can be expressed in the form m/n, where m and n are relatively prime positive integers. Find m + n.

Answer 1

In triangle MAN, the value of sin angle MAN is sqrt(2).

Step-by-step explanation:

In triangle MAN, we know that the side MA is equal to the diagonal of the square ABCD, which is equal to the length of the side of the square, let's call it s. Similarly, the side AN is also equal to s. The side MN is the length of the side of the square times the square root of 2, because the diagonal of a square is equal to the side length times the square root of 2.

So, MN = s * sqrt(2).

To find the value of sin angle MAN, we need to find the ratio of the length of the opposite side to the length of the hypotenuse in triangle MAN.

The opposite side is MN, which is s * sqrt(2), and the hypotenuse is MA or NA, which is s. So, sin angle MAN = (s * sqrt(2)) / s = sqrt(2).