Difference between revisions of "2020 AMC 10A Problems/Problem 12"
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Finally knowing that the medians divide the triangle into <math>6</math> sections of equal area, finding the area of <math>\triangle PUM</math> is enough. <math>\overline{PC} = \overline{MP} = 8</math>. | Finally knowing that the medians divide the triangle into <math>6</math> sections of equal area, finding the area of <math>\triangle PUM</math> is enough. <math>\overline{PC} = \overline{MP} = 8</math>. | ||
− | The area of <math>\triangle PUM = \frac{4\cdot8}{2}=16</math>. Multiplying this by <math>6</math> gives us <math>6\cdot16=\boxed{\textbf{(C) 96}</math> ~quacker88 | + | The area of <math>\triangle PUM = \frac{4\cdot8}{2}=16</math>. Multiplying this by <math>6</math> gives us <math>6\cdot16=\boxed{\textbf{(C) 96}}</math> ~quacker88 |
==Video Solution== | ==Video Solution== |
Revision as of 23:48, 31 January 2020
Contents
Problem
Triangle is isoceles with . Medians and are perpendicular to each other, and . What is the area of
Solution
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 (CHEATING)
Draw a to-scale diagram with your graph paper and straightedge. Measure the height and approximate the area.
Solution 3 (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 4
(using Solution 3 diagram)
, and . Since and are and , respectively.
By symmetry, . We can now do Pythagorean Theorem on .
Solution 5 (FASTEST)
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
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.