There are two flags on a flag pole. The upper flag is an isosceles triangle and the lower flag is a right triangle. Both flags have a shared length along the flag pole of 7 ft and the width of the lower flag is 3 ft. The angle of the point of the upper flag α is 16 degrees and the angle of the point of the lower flag β is 45 degrees. Find the length of the side of the upper flag that is not attached to the flag pole.

For the lower flag, the unkonwn angle is 45° since the sum of the three angles is 180°. we have that it is also an isoceles triangle, since it contains two equal angles. In a isoceles triangle, the lenght of the sides opposite to the equal angles is the same. So the length of the side of the lower flag attached to the pole is 3 ft, hence, the side of the upper flag attached to the pole is 4 ft.

Given that upper flag is an isoceles triangle, both angles that we don’t know are equal. Since the sum of the three angles is 180° and the angle in the point is 16°, the sum of te other two angles is 164°, thus the angle is 82°.

Using the law of sines, we can find the lenght of the of the upper flag that is not attached to the flag pole.

Let X be of the side of the upper flag that is not attached to the flag pole.

X / sin 82° = 4 / sin 16°

X = = 4 / sin 16°* sin 82°

X = 14.37 ft