2008 Mock ARML 1 Problems/Problem 3
In regular hexagon with side length , intersects at , and intersects at . Compute the length of .
Let be the foot of the perpendicular from to . Since is an inscribed angle with measure , it follows that is a , and and . Also, . Note that by ratio . Thus .
By the Pythagorean Theorem, . Thus .
We use coordinates. Consider the diagram in Solution 1. Let . By 30-60-90 triangle ratios, we can get that , and . In addition, and . The line is represented by . The line is represented by . The intersection () is then . We additionaly can get that . Then, by the distance formula, the length of is .
Solution 3 (30 Seconds)
Use the diagram in Solution 1. Notice that . The ratio is , since and . Letting , we see that . Then by the Pythagorean Theorem, .
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