Difference between revisions of "2023 AMC 8 Problems/Problem 24"
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Solution by [[User:ILoveMath31415926535|ILoveMath31415926535]] | Solution by [[User:ILoveMath31415926535|ILoveMath31415926535]] | ||
+ | |||
+ | ==Solution 4 (Slower)== | ||
+ | The answers are there on the bottom, so start with the middle one, <math>\boxed{\textbf{(C)}15}</math>. We find that we need a shorter length, so try <math>\boxed{\textbf{(B)}14.8}</math>. Still, we need a shorter answer, so we simply choose <math>\boxed{\textbf{(A)}14.6}</math> without trying it out. | ||
+ | |||
+ | ~SaxStreak | ||
==Video Solution 1 by OmegaLearn (Using Similarity)== | ==Video Solution 1 by OmegaLearn (Using Similarity)== |
Revision as of 08:52, 27 February 2023
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2 (Thorough)
- 4 Solution 3 (Faster)
- 5 Solution 4 (Slower)
- 6 Video Solution 1 by OmegaLearn (Using Similarity)
- 7 Video Solution 2 by SpreadTheMathLove(Using Area-Similarity Relaitionship)
- 8 Video Solution 3 by Magic Square (Using Similarity and Special Value)
- 9 Video Solution by Interstigation
- 10 Video Solution by WhyMath
- 11 See Also
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
Solution 3 (Faster)
Since the length of AC does not matter, we can assume the base of triangle ABC is . Therefore, the area of the trapezoid in the first diagram is
.
The area of the triangle in the second diagram is now .
Therefore, . Multiplying both sides by
, we get
. Subtracting
from both sides, we get
and
is
.
Solution by ILoveMath31415926535
Solution 4 (Slower)
The answers are there on the bottom, so start with the middle one, . We find that we need a shorter length, so try
. Still, we need a shorter answer, so we simply choose
without trying it out.
~SaxStreak
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
Video Solution by Interstigation
https://youtu.be/1bA7fD7Lg54?t=2454
Video Solution by WhyMath
~savannahsolver
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.