2020 AMC 10A Problems/Problem 12
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 (Pythagorean Theorem)
Let be the height. It passes through
because
is isosceles. It is well known that medians divide each other into segments of
ratio. From this, we have
and
. From right triangle
,
so
. Since
is a median,
. From right triangle
,
, which implies
. By symmetry
.
Applying the Pythagorean Theorem to 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
.
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 |
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