2002 AMC 8 Problems/Problem 20
Contents
Problem
The area of triangle is 8 square inches. Points and are midpoints of congruent segments and . Altitude bisects . The area (in square inches) of the shaded region is
Solution 1
The shaded region is a right trapezoid. Assume WLOG that . Then because the area of is equal to 8, the height of the triangle . Because the line is a midsegment, the top base of the trapezoid is . Also, divides in two, so the height of the trapezoid is . The bottom base is . The area of the shaded region is .
Solution 2
Since and are the midpoints of and , respectively, . Draw segments and . Drawing an altitude in an isoceles triangle splits the triangle into 2 congruent triangles and we also know that . is the line that connects the midpoints of two sides of a triangle together, which means that is parallel to and half in length of . Then . Since is parallel to , and is the transversal, Similarly, Then, by SAS, . Since corresponding parts of congruent triangles are congruent,. Since ACY and BCZ are now isosceles triangles, and Using the fact that is parallel to , and . Now . Draw an altitude through each of them such that each triangle is split into two congruent right triangles. Now there are a total of 8 congruent small triangles, each with area 1. The shaded area has three of these triangles, so it has area 3.
Basically the proof is to show . If you just look at the diagram you can easily see that the triangles are congruent and you would solve this a lot faster. Anyways, since those triangles are congruent, you can split each in half to find eight congruent triangles with area 1, and since the shaded region has three of these triangles, its area is .
Solution 3
We know the area of triangle is square inches. The area of a triangle can also be represented as or in this problem . By solving, we have
With SAS congruence, triangles and are congruent. Hence, triangle . (Let's say point is the intersection between line segments and .) We can find the area of the trapezoid by subtracting the area of triangle from .
We find the area of triangle by the formula- . is of from solution 1. The area of is .
Therefore, the area of the shaded area- trapezoid has area .
- sarah07
Solution 4 (Dummed down)
The area of triangle is square inches. You can turn the triangle into a rectangle by drawing a triangle on the left side with the hypotenuse of it being and another triangle on the right side, with its hypotenuse being . After drawing the square, you can cut it into squares. The area of the rectangle is square inches because the triangle on the left is half of and there's another triangle on the other side, equaling them to square inches. We will be focusing on the left side of the rectangle of because it includes the shaded region. It's split into equal squares. The area of this triangle is square inches because the total area of the rectangle is square inches and divided by is . There are sections, so you would do divided by in order to find the area of one square. That means that the area of the top right square is inches and because it's not needed, we will subtract from to get rid of it. If you turn around the paper, you notice that the square with half of the shaded region, and the square above it is the shaded region, except flipped and turned. Therefore, the remaining of the area of it which is should then be divided by in order to find the shaded region since the shaded region is equal to the other square and half a square. divided by is , so the answer is .
Solution 5 (Dividing into Congruent Triangles)
We draw the triangles as shown above. What seems like a half-square (or a corner of a square) indicates °. (This is not a random idea - the two original two triangles on the top inspires us to do so.)
Note that is perpendicular to ; is perpendicular to .
is perpendicular to , so because of corresponding angles theorem, . We are also told . Both and have a ° angle. Because of AAS Congruency (Angle-Angle-Side; °, , and side ), is congruent to . Because of symmetry, the same goes for and .
is congruent to , which is congruent to because of symmetry. So there are congruent triangles on the left side. There is also congruent triangles on the right side because of symmetry.
There are congruent triangles in all. We are told the area of the triangle is 8. , so each of the small congruent triangles is . The shaded area contains small triangles, so the area of the shaded section is .
- JoyfulSapling
Video Solution
https://www.youtube.com/watch?v=zwy5U5IQi88 ~David
See Also
2002 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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