2024 AMC 8 Problems/Problem 11
Contents
[hide]- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Solution 4
- 6 Video by MathTalks 😉
- 7 Video Solution by Math-X (First understand the problem!!!)
- 8 Video Solution by Central Valley Math Circle(Goes Through Full Thought Process)
- 9 Video Solution (A Clever Explanation You’ll Get Instantly)
- 10 Video Solution (easy to digest) by Power Solve
- 11 Video Solution by NiuniuMaths (Easy to understand!)
- 12 Video Solution 3 by SpreadTheMathLove
- 13 Video Solution by CosineMethod [🔥Fast and Easy🔥]
- 14 Video Solution by Interstigation
- 15 Video Solution by Daily Dose of Math (Certified, Simple, and Logical)
- 16 Video Solution by Dr. David
- 17 Video Solution by WhyMath
- 18 See Also
Problem
The coordinates of are
,
, and
, with
. The area of
is 12. What is the value of
?
Solution 1
Since the triangle has a base of , we can plug in that value as the base. Then, we can solve the equation for the height. Doing so gives us,
This means that
, so that means that we have to add 4 to the
-coordinate. So the answer is
Solution 2
By the Shoelace Theorem, has area
From the problem, this is equal to
. We now solve for y.
OR
OR
OR
However, since, as stated in the problem, , our only valid solution is
.
~ cxsmi
Solution 3
As in the figure, the triangle is determined by the vectors and
. Recall that the absolute value of the determinant of these vectors is the area of the parallelogram determined by those vectors; the triangle has half the area of that parallelogram. Then we must have that
. Expanding the determinants, we find that
or
. Solving each equation individually, we find that
or
. However, the problem states that
, so the only valid solution is
.
~ cxsmi (again!)
Solution 4
Draw a rectangle so that the hypotenuse is the diagonal of the rectangle. The base is 8 and the height is
Inside the rectangle, there are 2 extra right triangles along with the original triangle. The larger right triangle has an area of half the rectangle, so it has an area of
The smaller right triangle has a base of 2 and a height of
Subtracting the extra right triangles from the area of the rectangle, you get
Since the problem told us that the original triangle had an area of 12, you get the equation
So the answer to the problem is
Video by MathTalks 😉
https://youtu.be/qAwRUj2N46c?si=QDUY8ZUVFP29Eg4c
~rc1219
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/BaE00H2SHQM?si=qhPbhu8o5hamBrtb&t=2315
~Math-X
Video Solution by Central Valley Math Circle(Goes Through Full Thought Process)
~mr_mathman
Video Solution (A Clever Explanation You’ll Get Instantly)
https://youtu.be/5ZIFnqymdDQ?si=6FzUoSOA5moM-gDP&t=1191
~hsnacademy
Video Solution (easy to digest) by Power Solve
https://www.youtube.com/watch?v=2UIVXOB4f0o
Video Solution by NiuniuMaths (Easy to understand!)
https://www.youtube.com/watch?v=V-xN8Njd_Lc
~NiuniuMaths
Video Solution 3 by SpreadTheMathLove
https://www.youtube.com/watch?v=RRTxlduaDs8
Video Solution by CosineMethod [🔥Fast and Easy🔥]
https://www.youtube.com/watch?v=-64aBL-lEVg
Video Solution by Interstigation
https://youtu.be/ktzijuZtDas&t=1063
Video Solution by Daily Dose of Math (Certified, Simple, and Logical)
~Thesmartgreekmathdude
Video Solution by Dr. David
Video Solution by WhyMath
See Also
2024 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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 AJHSME/AMC 8 Problems and Solutions |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.