2019 AMC 10A Problems/Problem 13
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
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
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
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 .
Solution 5 (CHEATING)
If you can't see how to solve it, you could simply draw an accurate diagram and measure the angle using a protractor as 110 - .
is isosceles so . Since is a diameter, . Quadrilateral is cyclic since . Therefore
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