First, let's start by splitting it up into pieces we can identify. The easiest to spot is the semicircle on the lower left side (going from A to B to C). The remaining pentagon (ACDEF) can be cut from A to D to form a triangle (ΔACD) and a quadrilateral (ADEF). If we add these individual areas, we'll get the area of the entire figure. Let's start by calculating the area of the semicircle, A_{C}. Since its diameter goes from A to C, it's clear that the radius of the semicircle is 3. We can multiply the area formula for a circle by ½ to find the area of the semicircle.
A_{C} = 4.5π ≈ 14.1 units^{2} It's clear that ΔACD is a right triangle, with legs of lengths 3 – 0 = 3 and 2 – (-4) = 6. We can plug those numbers in as the base and height and find the area. A_{T} = ½bh A_{T} = ½(3)(6) A_{T} = 9 units^{2} Now for that quadrilateral. Before we can do anything, let's find out what type of quadrilateral it really is. Calculating side lengths is a good way to do this. Go, go, distance formula! We have two pairs of congruent sides. What we have here is a parallelogram, so we know to use A = bh to find the area. Only one slight problem: what's the base and what's the height? We know that the height has to be perpendicular to the base, so let's start there. We'll set our base to DE, which has a slope of 4. Our height should have a slope of (since negative reciprocal slopes mean perpendicular lines). Calculating the height of this thing might be trickier than you'd expect, so we'd suggest setting up a system of linear equations. For instance, the intersection of lines and y = 4x – 16 will give us the point on a line perpendicular to DE that goes through (0, 2). Using the distance formula between that point, , and (0, 2) should give us the height. Now we can use A = bh to find the area of the parallelogram. Rather than plugging in , we can estimate it at 16.49. A_{P} = bh A_{P} = 16.49 × 4.37 A_{P} ≈ 72.1 units^{2} We're still not done. Adding the pieces together will solve the last piece of the puzzle. A = A_{P} + A_{T} + A_{C} A = 72.1 units^{2} + 9 units^{2} + 14.1 units^{2} A = 95.2 unis^{2} |