BTGmoderatorDC wrote:
A sheet of paper ABDE is a 12-by-18-inch rectangle, as shown in Figure 1. The sheet is then folded along the segment CF so that points A and D coincide after the paper is folded, as shown in Figure 2 (The shaded area represents a portion of the back side of the paper, not visible in Figure 1). What is the area, in square inches, of the shaded triangle shown?
A) 72
B) 78
C) 84
D) 96
E) 108
OA
B
Source: Manhattan Prep
We can let BC = x and thus CD = 18 - x. We see that triangle DBC (in figure 2) is a right triangle and thus we have:
(DB)^2 + (BC)^2 = (CD)^2
12^2 + x^2 = (18 - x)^2
144 + x^2 = 324 - 36x + x^2
36x = 180
x = 5
Likewise, if we let FE = y and thus AF = 18 - y. We see that triangle AEF (in figure 2) is a right triangle. Since side AE of triangle AEF is now same as DE, AE = 12 and we have:
(AE)^2 + (FE)^2 = (AF)^2
12^2 + y^2 = (18 - y)^2
This is equivalent to the equation above, so y = 5.
Now we can argue that:
Area of triangle DBC + Area of triangle AEF + 2(Area of triangle CAF) = Area of rectangle ABDE
(½)(12)(5) + (½)(12)(5) + 2(Area of triangle CAF) = 12(18)
30 + 30 + 2(Area of triangle CAF) = 216
2(Area of triangle CAF) = 156
Area of triangle CAF = 78
But the area of triangle CAF is the area of the shaded region, so the area of the shaded region is 78.
Answer: B
