2021 Fall AMC 10B Problems/Problem 15
Contents
[hide]Problem
In square , points
and
lie on
and
, respectively. Segments
and
intersect at right angles at
, with
and
. What is the area of the square?
Solution 1
Note that Then, it follows that
Thus,
Define
to be the length of side
then
Because
is the altitude of the triangle, we can use the property that
Substituting the given lengths, we have
Solving, gives
and
We eliminate the possibilty of
because
Thus, the side lengnth of the square, by Pythagorean Theorem, is
Thus, the area of the sqaure is
Thus, the answer is
~NH14
Solution 2
As above, note that , which means that
. In addition, note that
is the altitude of a right triangle to its hypotenuse, so
. Let the side length of the square be
; using similarity side ratios of
to
, we get
. Note that
by the Pythagorean theorem, so we can use the expansion
to produce two equations and two variables;
We want , so we want to find
. Subtracting the first equation from the second, we get
Then =
~KingRavi
Solution 3
We have that Thus,
. Now, let the side length of the square be
Then, by the Pythagorean theorem,
Plugging all of this information in, we get
Simplifying gives
Squaring both sides gives
We now set
and get the equation
From here, notice we want to solve for
, as it is precisely
or the area of the square. So we use the Quadratic formula, and though it may seem bashy, we hope for a nice cancellation of terms.
It seems scary, but factoring
from the square root gives us
giving us the solutions
We instantly see that
is way too small to be an area of this square (
isn't even an answer choice, so you can skip this step if out of time) because then the side length would be
and then, even the largest line you can draw inside the square (the diagonal) is
which is less than
(line
) And thus,
must be
, and our answer is
~wamofan
Solution 4 (Point-line distance formula)
Denote . Now tilt your head to the right and view
and
as the origin,
-axis and
-axis, respectively. In particular, we have points
. Note that side length of the square
is
. Also equation of line
is
Because the distance from
to line
is also the side length
, we can apply the point-line distance formula to get
which reduces to
. Since
is positive, the last equations factors as
. Now judging from the figure, we learn that
. So
.
Therefore, the area of the square
is
. Choose
.
~VensL.
See Also
2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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.