A parallelogram has perimeter 40 and altitudes of length 4 and 7. Find the sine of the smaller vertex angle.
The parallelogram has a shorter side (let's call it 'x') and a longer side (which is '20-x').
The altitude of length 4 goes to the longer side. Using the area of a parallelogram to be its side times its altitude,
the area of the parallelogram is 4·(20 - x).
Using the shorter side (x) and its altitude (7) to find the area, we get 7·x.
Therefore: 4·(20 - x) = 7·x
80 - 4x = 7x
80 = 11x
x = 80/11
To find the sine of the smaller vertex angle: sine(angle) = 4 / (80/11) = 44/80 = 11/20