Difference between revisions of "2020 AMC 10A Problems/Problem 20"
m (→Solution 2 (Pro Guessing Strats)) |
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==Solution 3 (coordinates)== | ==Solution 3 (coordinates)== | ||
+ | <asy> | ||
+ | draw((10,30)--(10,0)--(-8,-6)--(-10,0)--(10,30)); | ||
+ | draw(circle((0,0),10)); | ||
+ | label("D",(10,30),N); | ||
+ | label("C",(10,0),E); | ||
+ | label("B",(-8,-6),S); | ||
+ | label("A",(-10,0),W); | ||
+ | </asy> | ||
Let the points be <math>A(-10,0)</math>, <math>\:B(x,y)</math>, <math>\:C(10,0)</math>, <math>\:D(10,30)</math>,and <math>\:E(-5,0)</math>, respectively. Since <math>B</math> lies on line <math>DE</math>, we know that <math>y=2x+10</math>. Furthermore, since <math>\angle{ABC}=90^\circ</math>, <math>B</math> lies on the circle with diameter <math>AC</math>, so <math>x^2+y^2=100</math>. Solving for <math>x</math> and <math>y</math> with these equations, we get the solutions <math>(0,0)</math> and <math>(-8,-6)</math>. We immediately discard the <math>(0,0)</math> solution as <math>y</math> should be negative. Thus, we conclude that <math>[ABCD]=[ACD]+[ABC]=\frac{20\cdot30}{2}+\frac{20\cdot6}{2}=\boxed{\textbf{(D)}\:360}</math>. | Let the points be <math>A(-10,0)</math>, <math>\:B(x,y)</math>, <math>\:C(10,0)</math>, <math>\:D(10,30)</math>,and <math>\:E(-5,0)</math>, respectively. Since <math>B</math> lies on line <math>DE</math>, we know that <math>y=2x+10</math>. Furthermore, since <math>\angle{ABC}=90^\circ</math>, <math>B</math> lies on the circle with diameter <math>AC</math>, so <math>x^2+y^2=100</math>. Solving for <math>x</math> and <math>y</math> with these equations, we get the solutions <math>(0,0)</math> and <math>(-8,-6)</math>. We immediately discard the <math>(0,0)</math> solution as <math>y</math> should be negative. Thus, we conclude that <math>[ABCD]=[ACD]+[ABC]=\frac{20\cdot30}{2}+\frac{20\cdot6}{2}=\boxed{\textbf{(D)}\:360}</math>. | ||
Revision as of 11:21, 1 February 2020
Contents
Problem
Quadrilateral satisfies
and
Diagonals
and
intersect at point
and
What is the area of quadrilateral
Solution 1
It's crucial to draw a good diagram for this one. Since and
, we get
. Now we need to find
to get the area of the whole quadrilateral. Drop an altitude from
to
and call the point of intersection
. Let
. Since
, then
. By dropping this altitude, we can also see two similar triangles,
and
. Since
is
, and
, we get that
. Now, if we redraw another diagram just of
, we get that
. Now expanding, simplifying, and dividing by the GCF, we get
. This factors to
. Since lengths cannot be negative,
. Since
,
. So
(I'm very sorry if you're a visual learner)
~Ultraman
Solution 2 (Pro Guessing Strats)
We know that the big triangle has area 300. Use the answer choices which would mean that the area of the little triangle is a multiple of 10. Thus the product of the legs is a multiple of 20. Guess that the legs are equal to and
, and because the hypotenuse is 20 we get
. Testing small numbers, we get that when
and
,
is indeed a square. The area of the triangle is thus 60, so the answer is
.
~tigershark22
Solution 3 (coordinates)
Let the points be
,
,
,
,and
, respectively. Since
lies on line
, we know that
. Furthermore, since
,
lies on the circle with diameter
, so
. Solving for
and
with these equations, we get the solutions
and
. We immediately discard the
solution as
should be negative. Thus, we conclude that
.
Video Solution
~IceMatrix
See Also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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.