2019 AMC 10A Problems/Problem 7
- The following problem is from both the 2019 AMC 10A #7 and 2019 AMC 12A #5, so both problems redirect to this page.
Problem
Two lines with slopes and intersect at . What is the area of the triangle enclosed by these two lines and the line
Solution 1
The two lines are and , which intersect the third line at and . So we have an isosceles triangle with base and height .
Solution 2
Like in Solution 1, let's first calculate the slope-intercept form of all three lines: and implies so , while implies so . Also, implies . Thus the lines are and . Now we find the intersections between each of the lines with , which are and . Applying the Shoelace Theorem, we can find that the solution is
Solution 3
Like the other solutions, solve the systems to see that the triangles two other points are at and . The apply Heron's Formula. The semi-perimeter will be . The area then reduces nicely to a difference of squares, making it
Solution 4
Using the methods above to find the coordinates, we get , , and . Using the Shoelace Theorem, we find that it is equal to . Because the Shoelace Theorem tells us to find the half of that sum, we get
--Claire (clara32356)
See Also
2019 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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 |
2019 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 4 |
Followed by Problem 6 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.