The window shown is the shape of a semicircle with a radius of 6 feet. the distance from f to e is 3 feet and the measure of = 45°. find the area of the glass in region bcih, rounded to the nearest square foot.

11 ft^2

Step-by-step explanation:

In the first figure attached, a graphical representation of the problem can be seen.

The area of the glass in region BCIH is equivalent to 1/8 of the "ring" area formed by the subtraction of the smaller circle to the bigger circle (see the second figure attached).

The number 1/8 comes from knowing that a circle form an angle of 360° and the angle between segments GC and GB is 45°, then, the area of the region of interest is 45°/360° = 1/8 respect the entire "ring".

The segment GF plus segment FE is equal to the radius of the bigger circle, which is 6 ft . Therefore,

GF + FE = 6 ft

GF = 6 ft - 3 ft

GF = 3 ft

"Ring" area is equal to the subtraction of the smaller circle area to the bigger circle area. Mathematically:

Ring area = pi*(6 ft)^2 - pi*(3 ft)^2

Ring area = 84.78 ft^2

Area of the glass in region BCIH = (1/8)*84.78 ft^2 = 11 ft^2

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4