2004 AMC 12B Problems/Problem 14
In , , , and . Points and lie on and , respectively, with . Points and are on so that and are perpendicular to . What is the area of pentagon ?
The triangle is clearly a right triangle, its area is . If we knew the areas of triangles and , we could subtract them to get the area of the pentagon.
Draw the height from onto . As and the area is , we get . The situation is shown in the picture below:
Now note that the triangles , , , and all have the same angles and therefore they are similar. We already know some of their sides, and we will use this information to compute their areas. Note that if two polygons are similar with ratio , their areas have ratio . We will use this fact repeatedly. Below we will use to denote the area of the triangle .
We have , hence .
Also, , hence .
Now for the smaller triangles:
We know that , hence .
Similarly, , hence .
Finally, the area of the pentagon is .
Split the pentagon along a different diagonal as follows:
The area of the pentagon is then the sum of the areas of the resulting right triangle and trapezoid. As before, triangles , , and are all similar.
Since , and . Since , and .
The trapezoid's height is therefore , and its area is .
Triangle has area , and the total area is .
Right triangle has area .
We are given that and , so . Right triangles and are similar, with hypotenuses 1 and 13, respectively, so triangle has area Also, we are given that and , so . Right triangles and are similar, with hypotenuses 8 and 13, respectively, so triangle has area Therefore, the area of pentagon is
Because triangle ABC, triangle NBK, and triangle AMJ are similar right triangles whose hypotenuses are in the ratio 13 : 8 : 1, their areas are in the ratio 169 : 64 : 1. The area of triangle ABC is 1/2 (12)(5) = 30, so the areas of triangle NBK and triangle AMJ are (64/169) (30) and (1/169)(30), respectively. Thus the area of pentagon CMJKN is (1 − 64/169 - 1/169)(30)=
Credit to http://billingswest.billings.k12.mt.us/math/AMC%201012/AMC%2012%20work%20sheets/2004%20AMC%2012B%20ws-15.pdf for Solution 3.
We fake-solve. Draw a perpendicular line to side that goes through . Similarly, draw a perpendicular line to side that goes through . Let them intersect at . It follows that quadrilateral is a square with side and thus area . Now, draw a perpendicular from to and let them intersect at . There is a obvious pair of congruent triangles. Fill in the gap. Hence, we see that the area of the pentagon is definitely greater than . How much greater? Well, if we let the intersection of and be , we can approximate that has an area greater than . So we place bets on which is correct!
This would be much easier to show and more convincing with a diagram but idk how to upload GeoGebra to AoPSwiki.
Solution by franzliszt
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