Difference between revisions of "2021 Fall AMC 12B Problems/Problem 8"
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~Steven Chen (www.professorchenedu.com) | ~Steven Chen (www.professorchenedu.com) | ||
+ | ==Video Solution by TheBeautyofMath== | ||
+ | https://youtu.be/kuZXQYHycdk?t=795 | ||
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+ | ~IceMatrix | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2021 Fall|ab=B|num-a=9|num-b=7}} | {{AMC12 box|year=2021 Fall|ab=B|num-a=9|num-b=7}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 00:39, 30 December 2022
Problem
The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?
Solution 1 (Area)
Let the lengths of the two congruent sides of the triangle be , then the product desired is
.
Notice that the product of the base and twice the height is times the area of the triangle.
Set the vertex angle to be , we derive the equation:
As the triangle is obtuse, only. We get
~Wilhelm Z
Solution 2
Denote by the length of each congruent side. Denote by
the degree measure of each acute angle.
Denote by
the degree measure of the obtuse angle.
Hence, this problem tells us the following relationship:
Hence,
Hence, .
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
Video Solution by TheBeautyofMath
https://youtu.be/kuZXQYHycdk?t=795
~IceMatrix
See Also
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.