2017 AMC 12B Problems/Problem 24
Quadrilateral has right angles at and , , and . There is a point in the interior of such that and the area of is times the area of . What is ?
Let , , and . Note that . By the Pythagorean Theorem, . Since , the ratios of side lengths must be equal. Since , and . Let F be a point on such that is an altitude of triangle . Note that . Therefore, and . Since and form altitudes of triangles and , respectively, the areas of these triangles can be calculated. Additionally, the area of triangle can be calculated, as it is a right triangle. Solving for each of these yields: Therefore, the answer is
Draw line through , with on and on , . WLOG let , , . By weighted average .
Meanwhile, . This follows from comparing the ratios of triangle DEG to CFE and triangle AEG to FEB, both pairs in which the two triangles share a height perpendicular to FG, and have base ratio .
. We obtain , namely .
The rest is the same as Solution 1.
Let . Then from the similar triangles condition, we compute and . Hence, the -coordinate of is just . Since lies on the unit circle, we can compute the coordinate as . By Shoelace, we want Factoring out denominators and expanding by minors, this is equivalent to This factors as , so and so the answer is .
Let where . Because . Notice that the diagonals are perpendicular with slopes of and . Let the intersection of and be , then . However, because is a trapezoid, and share the same area, therefore is the reflection of over the perpendicular bisector of , which is . We use the linear equations of the diagonals, , to find the coordinates of . The y-coordinate of is simply The area of is . We apply shoelace theorem to solve for the area of . The coordinates of the triangle are , so the area is
Finally, we use the property that the ratio of areas equals
1) is the most relevant answer choice because it shares numbers with the givens of the problem.
2) It's a very good guess to replace finding the area of triangle AED with the area of the triangle DAF, where F is the projection of D onto AB(then find the closest answer choice).
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