Difference between revisions of "2020 AMC 10A Problems/Problem 12"
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So <math>UP = VP = 4</math>, <math>MP = PC = 8</math>. | So <math>UP = VP = 4</math>, <math>MP = PC = 8</math>. | ||
− | With that, we can see that <math> | + | With that, we can see that <math>[\triangle UPM] = 16</math>, and the area of trapezoid <math>MUVC</math> is 72. |
− | As said in solution 1, <math> | + | As said in solution 1, <math>[\triangle AMC] = 72 / \frac{3}{4} = \boxed{\textbf{(C) } 96}</math>. |
-QuadraticFunctions, solution 1 by ??? | -QuadraticFunctions, solution 1 by ??? |
Revision as of 15:49, 4 February 2020
Contents
Problem
Triangle is isoceles with . Medians and are perpendicular to each other, and . What is the area of
Solution 1
Since quadrilateral has perpendicular diagonals, its area can be found as half of the product of the length of the diagonals. Also note that has the area of triangle by similarity, so Thus,
Solution 2 (Trapezoid)
We know that , and since the ratios of its sides are , the ratio of of their areas is .
If is the area of , then trapezoid is the area of .
Let's call the intersection of and . Let . Then . Since , and are heights of triangles and , respectively. Both of these triangles have base .
Area of
Area of
Adding these two gives us the area of trapezoid , which is .
This is of the triangle, so the area of the triangle is ~quacker88, diagram by programjames1
Solution 3 (Medians)
Draw median .
Since we know that all medians of a triangle intersect at the incenter, we know that passes through point . We also know that medians of a triangle divide each other into segments of ratio . Knowing this, we can see that , and since the two segments sum to , and are and , respectively.
Finally knowing that the medians divide the triangle into sections of equal area, finding the area of is enough. .
The area of . Multiplying this by gives us
~quacker88
Solution 4 (Triangles)
We know that , , so .
As , we can see that and with a side ratio of .
So , .
With that, we can see that , and the area of trapezoid is 72.
As said in solution 1, .
-QuadraticFunctions, solution 1 by ???
Solution 5 (Only Pythagorean Theorem)
Let be the height. Since medians divide each other into a ratio, and the medians have length 12, we have and . From right triangle , so . Since is a median, . From right triangle , which implies . By symmetry .
Applying the Pythagorean Theorem to right triangle gives , so . Then the area of is
Solution 6 (Drawing)
(NOT recommended) Transfer the given diagram, which happens to be to scale, onto a piece of a graph paper. Counting the boxes should give a reliable result since the answer choices are relatively far apart. -Lingjun
Solution 7
Given a triangle with perpendicular medians with lengths and , the area will be .
Solution 8 (Fastest)
Connect the line segment and it's easy to see quadrilateral has an area of the product of its diagonals divided by which is . Now, solving for triangle could be an option, but the drawing shows the area of will be less than the quadrilateral meaning the the area of is less than but greater than , leaving only one possible answer choice, .
-Rohan S.
Video Solution
~IceMatrix
See Also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.