Difference between revisions of "2023 AMC 8 Problems/Problem 24"
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− | We can call the length of AC as <math>x</math>. Therefore, the length of the base of the triangle with height <math>11</math> is <math>11/h = a/x</math>. Therefore, the base of the smaller triangle is <math>11x/h</math>. We find that the area of the trapezoid is <math>( | + | We can call the length of AC as <math>x</math>. Therefore, the length of the base of the triangle with height <math>11</math> is <math>11/h = a/x</math>. Therefore, the base of the smaller triangle is <math>11x/h</math>. We find that the area of the trapezoid is <math>(hx)/2 - 11^2x/2h</math>. |
− | Using similar triangles once again, we find that the base of the shaded triangle is <math>(h-5)/h = b/x</math>. Therefore, the area is <math>(h-5)(hx-5x)/h</math>. | + | Using similar triangles once again, we find that the base of the shaded triangle is <math>(h-5)/h = b/x</math>. Therefore, the area is <math>(h-5)(hx-5x)/h</math>. |
+ | |||
+ | Since the areas are the same, we find that <math>(hx)/2 - 121x/2h = (h-5)(hx-5x)/h</math>. Multiplying each side by <math>2h</math>, we get <math>h^2x - 121x = h^2x - 5hx - 5hx + 25x</math>. Therefore, we can subtract <math>25x + h^2x</math> from both sides, and get <math>-146x = -10hx</math>. Finally, we divide both sides by <math>-x</math> and get <math>10h = 146x</math>. <math>h</math> is <math>\boxed{\textbf{(A)}14.6}</math>. | ||
+ | |||
+ | Solution by [[ILoveMath31415926535|ILoveMath31415926535]] | ||
==Video Solution 1 by OmegaLearn (Using Similarity)== | ==Video Solution 1 by OmegaLearn (Using Similarity)== |
Revision as of 13:01, 25 January 2023
Contents
Problem
Isosceles has equal side lengths
and
. In the figure below, segments are drawn parallel to
so that the shaded portions of
have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height of
of
?
(note: diagrams are not necessarily drawn to scale)
Solution 1
First, we notice that the smaller isosceles triangles are similar to the larger isosceles triangles. We can find that the area of the gray area in the first triangle is . Similarly, we can find that the area of the gray part in the second triangle is
. These areas are equal, so
. Simplifying yields
so
.
~MathFun1000 (~edits apex304)
Solution 2 (thorough)
We can call the length of AC as . Therefore, the length of the base of the triangle with height
is
. Therefore, the base of the smaller triangle is
. We find that the area of the trapezoid is
.
Using similar triangles once again, we find that the base of the shaded triangle is . Therefore, the area is
.
Since the areas are the same, we find that . Multiplying each side by
, we get
. Therefore, we can subtract
from both sides, and get
. Finally, we divide both sides by
and get
.
is
.
Solution by ILoveMath31415926535
Video Solution 1 by OmegaLearn (Using Similarity)
Video Solution 2 by SpreadTheMathLove(Using Area-Similarity Relaitionship)
https://www.youtube.com/watch?v=GTlkTwxSxgo
Video Solution 3 by Magic Square (Using Similarity and Special Value)
https://www.youtube.com/watch?v=-N46BeEKaCQ&t=1569s
See Also
2023 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.