2014 AMC 10B Problems/Problem 21
Problem
Trapezoid has parallel sides of length and of length . The other two sides are of lengths and . The angles and are acute. What is the length of the shorter diagonal of ?
Solution
In the diagram, . Denote and . In right triangle , we have from the Pythagorean theorem: . Note that since , we have . Using the Pythagorean theorem in right triangle , we have .
We isolate the term in both equations, getting $\begin{align*}h^2 &= 100-x^2
\\h^2 &= 196-(12-x)^2\end{align}$ (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.).
Setting these equal, we have . Now, we can determine that . The two diagonals are and . Using the Pythagorean theorem again on and , we can find these lengths to be and . Obviously, is the shorter length, and thus the answer is
See Also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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All AMC 10 Problems and Solutions |
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