Problem number 30. In the pentagon ABCDE, angle A equals angle B equals 90 degrees, AE equals BC, and ED equals DC. 4 points are marked on AB, dividing it into 5 equal parts. Then perpendiculars are drawn through these points, as shown in the diagram. The dark shaded region has an area of 13 cm2, and the light shaded region has an area of 10 cm2. What is the area in cm2 of the entire pentagon? In this problem, we need to divide this pentagon up into many smaller shapes and find the areas of these smaller shapes. For example, let's draw parallel lines FG and JK as shown, where these perpendiculars hit the slanted sides of the pentagon. Since we know that angle A equals angle B equals 90 degrees, these lines FG and JK are parallel to the side EC. Now, let's find the areas of some of these shapes, for example the yellow quadrilateral. We know that the area of this light grey shaded region is 10, and the area of the dark grey shaded region is 13, so the area of this yellow quadrilateral is 3. Now, let's divide this yellow quadrilateral into several pieces as shown in the bottom left figure. Now, we can see that this yellow quadrilateral has been divided into 6 triangles, and so the area of each of these triangles is a half. Now, we can notice that this dark green triangle is made up of 4 of these small yellow triangles, and so that means that the area of each of these dark green triangles is 2. Finally, we can notice that each of these dark green triangles is the area of one of these light green rectangles divided by 2, since you can put two of them side by side to form the light green rectangle. That means that each of the light green rectangles has area of 4. Now, if we just look at the light grey shaded region in the original picture, we can see it's made up of a dark green triangle, a light green rectangle, and a cyan rectangle. We know that the area of the dark green triangle is 2, and the area of the light green rectangle is 4. That means that the area of the cyan rectangle is 10 minus 2 minus 4 equals to 4. Now, we can look at the region that's in the leftmost trapezoid in the original figure, which is essentially the white trapezoid, the smallest one in the original figure. Its area is just 2, which is the dark green triangle, plus 4, which is the area of the cyan rectangle, or 6. That means that we can label the area of that smallest trapezoid with 6. We know that the area of the light grey region is 10, and the area of the dark grey region is 13. We also know that this pentagon is symmetric, because all of the sides AE equals BC and ED equals DC. That means that this pentagon is symmetric about a perpendicular through D, and that means that the other two trapezoids have areas of 10 and 6. So, the area of the entire pentagon is equal to 6 plus 10 plus 13 plus 10 plus 6, which is equal to 45. So the answer is A, 45 centimeters squared.

pentagon

geometry

area calculation

symmetry

trapezoid