Difference between revisions of "2019 AMC 10A Problems/Problem 13"
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+ | ==Solution 4== | ||
+ | <asy> unitsize(40);draw((-1,0)--(1,0)--(0,2.75)--cycle);draw(circumcircle((-1,0),(0,0),(0,2.75)));label("$A$",(1,0),SE);label("$C$",(0,2.75),N);label("$B$",(-1,0),SW);label("$E$",(0,0),S);label("$D$",(0.77,0.64),E);draw((0,0)--(0,2.75));draw((-1,0)--(0.77,0.64));</asy> | ||
+ | |||
+ | Notice that if <BEC is 90 degrees, then <BCE and <ACE must be 20 degrees. Using cyclic quadrilateral properties (or the properties of a subtended arc), we can find that <EBD ~ <ECD = 20 degrees. Thus <CBF is 70 - 20 = 50 degrees, and so <BFC is 180 - 20 - 50 = 110 degrees, which is D. | ||
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==See Also== | ==See Also== |
Revision as of 21:01, 11 February 2019
Problem
Let be an isosceles triangle with and . Construct the circle with diameter , and let and be the other intersection points of the circle with the sides and , respectively. Let be the intersection of the diagonals of the quadrilateral . What is the degree measure of
Solution 1
Drawing it out, we see and are right angles, as they are inscribed in a semicircle. Using the fact that it is an isosceles triangle, we find . We can find and by the triangle angle sum on and .
Then, we take triangle , and find
~Argonauts16 (Diagram by Brendanb4321)
Solution 2
Alternatively, we could have used similar triangles. We start similarly to Solution 1.
Drawing it out, we see and are right angles, as they are inscribed in a semicircle. Therefore,
So, by AA Similarity, since and .
Thus, we know:
Finally, we know:
~ alleycat (Diagram by Brendanb4321)
Solution 3
Through the property of angles formed by intersecting chords, we find that
Through the Outside Angles Theorem, we find that
Adding the two equations gives us
Since is the diameter, and because is isosceles and , . Thus
~mn28407
Solution 4
Notice that if <BEC is 90 degrees, then <BCE and <ACE must be 20 degrees. Using cyclic quadrilateral properties (or the properties of a subtended arc), we can find that <EBD ~ <ECD = 20 degrees. Thus <CBF is 70 - 20 = 50 degrees, and so <BFC is 180 - 20 - 50 = 110 degrees, which is D.
See Also
2019 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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 |
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