19. In the figure above, AB is the diameter of the circle, and AC = BC. What is the area of the shaded region?
     
    (A) 4π – 2
    (B) 2π – 1
    (C) π
    (D) π – 1
    (E) π – 2

Answer and explanation after the jump…

As is usually the case with shaded region problems, the easiest way (and in this case, the only way) to solve for the shaded region is to solve for the regions around it first. The relationship we want to keep in mind is:

In this particular case, we’re going to have to do a little legwork to figure out what out “whole” is before we get down to business.

Let’s start by dropping a vertical from the top of our isosceles triangle (and noting that in doing so, we’re drawing a radius, so it’s got a length of 2):

That vertical is of course perpendicular to AB, and creates a right angle that nicely frames the area we’re looking to solve for. So the Area_whole we’re looking for here is actually only the part of the circle marked off by that right angle. Since a circle has 360 degrees of arc and we’re only dealing with 90 of them, we’re dealing with one fourth of the circle.

Area_circle = πr²
Area_circle = π(2²)
Area_circle = 4π

So the area of the sector we care about is simply one fourth of that, or π:

Area_whole = π

Now we just need to find the area of the unshaded part (the right triangle we created, in red):

Area_unshaded = 1/2 (2)(2)
Area_unshaded = 2

So the area of our shaded region must be…

Area_shaded = π – 2

That’s answer choice (E). 

One more note about this one: since the diagram is drawn to scale, it’s possible (and wise) to use your guesstimation skills once you’ve found an answer (or even to eliminate answers before you do much calculating). Which is to say: when you know the area of the triangle is 2, does it make sense that the shaded region is about 1.14159? I’d say, by eyeballing it, that yeah, it does. It wouldn’t have made sense, though, had we made a math mistake somehow and ended up with a different choice, like (D) π – 1. To pick that answer would be to say (insanely) that the shaded region is as big as the right triangle we made by dropping that vertical. That’s crazy talk.