Difference between revisions of "2019 AMC 10A Problems/Problem 13"
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==Solution 4== | ==Solution 4== | ||
− | Notice that if <math>\angle BEC = 90^{\circ}</math>, then <math>\angle | + | Notice that if <math>\angle BEC = 90^{\circ}</math>, then <math>\angle BCE</math> and <math>\angle ACE</math> must be <math>20^{\circ}</math>. Using cyclic quadrilateral properties (or the properties of a subtended arc), we can find that <math>\angle EBD \cong \angle ECD = 20^{\circ}</math>. Thus <math>\angle CBF = 70 - 20 = 50^{\circ}</math>, and so <math>\angle BFC = 180 - 20 - 50 = 110^{\circ}</math>, which is <math>\boxed{\textbf{(D)}}</math>. |
''Note'': As in many elementary geometry problems, if you can't see how to solve it, you could simply draw an accurate diagram and measure the angle using a protractor as <math>110^{\circ}</math>. | ''Note'': As in many elementary geometry problems, if you can't see how to solve it, you could simply draw an accurate diagram and measure the angle using a protractor as <math>110^{\circ}</math>. |
Revision as of 21:49, 5 January 2020
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
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 deduce
Solution 3 (outside angles)
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 , we have . Thus
Solution 4
Notice that if , then and must be . Using cyclic quadrilateral properties (or the properties of a subtended arc), we can find that . Thus , and so , which is .
Note: As in many elementary geometry problems, if you can't see how to solve it, you could simply draw an accurate diagram and measure the angle using a protractor as .
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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