2009 AMC 12B Problems/Problem 14
- The following problem is from both the 2009 AMC 10B #17 and 2009 AMC 12B #14, so both problems redirect to this page.
Contents
Problem
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from to , divides the entire region into two regions of equal area. What is ?
Solutions
Solution 1
For the shaded area is at most , which is too little. Hence , and therefore the point is indeed inside the shaded part, as shown in the picture.
Then the area of the shaded part is one less than the area of the triangle with vertices , , and . The area of the entire triangle is , therefore the area of the shaded part is .
The entire figure has area , hence we want the shaded part to have area . Solving for , we get . The answer is .
Solution 2
The unit square is of area 1, so the five unit squares have area 5. Therefore the shaded space must occupy 2.5. The missing unit square is of area 1, and if reconstituted the original triangle would be of area 3.5. It can then be inferred: .
, so .
. .
Solution 3
The shaded space of the object can become a triangle by adding a unit square to its bottom. This triangle has a base length of and a height of . The area of this triangle region is now (using the formula for a triangle). But, remember that we have to subtract the area of the extra unit square from this area to get the area of the shaded region.
If you add 3 unit squares to the top of the unshaded portion, the figure is now a right trapezoid. Using the formula for a trapezoid, you can derive that its area is . After you subtract the area of the 3 extra unit squares, we have derived the area of the unshaded portion.
Since these areas are both equal, we set them equal to each other and solve for . . is now solved to be . .
~Solution by athens2016
Solution 4 (Brute Force)
We are looking for the area of the shaded region to be . We start by testing . The area of the shaded region would be when . This does not match our wanted answer.
We try next. The area of the shaded region would be when . This also does not match our desired answer.
We then try . The area of the shaded region would be when . Therefore, our answer is .
Solution 5
Use the shoelace theorem to get the area of the unshaded portion in terms of a. Divide this area by the whole area which is 5. This fraction should be equal to . You can easily solve for the answer from there. -mewto
See Also
2009 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2009 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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