2020 AMC 10A Problems/Problem 12
Contents
Problem
Triangle is isosceles 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 centroid, 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.
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.
Solution 9
Connect , and let
be the point where
intersects
.
because all medians of a triangle intersect at one point, which in this case is
.
because the point at which all medians intersect divides the medians into segments of ratio
, so
and similarly
. We apply the Pythagorean Theorem to triangle
and get
. The area of triangle
is
, and that must equal to
, so
.
, so
. The area of triangle
is equal to
.
-SmileKat32
Video Solution 1
Education, The Study of Everything
Video Solution 2
~IceMatrix
Video Solution 3
~savannahsolver
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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