2009 AMC 10A Problems/Problem 17
Rectangle has and . Segment is constructed through so that is perpendicular to , and and lie on and , respectively. What is ?
The situation is shown in the picture below.
From the Pythagorean theorem we have .
Triangle is similar to , as they have the same angles. Segment is perpendicular to , meaning that angle and are right angles and congruent. Also, angle is a right angle. Because it is a rectangle, angle is congruent to and angle is also a right angle. By the transitive property:
Next, because every triangle has a degree measure of 180, angle and angle are congruent.
Hence , and therefore .
Also triangle is similar to . Hence , and therefore .
We then have .
Since is the altitude from to , we can use the equation .
Looking at the angles, we see that triangle is similar to . Because of this, . From the given information and the Pythagorean theorem, , , and . Solving gives .
We can use the above formula to solve for . . Solve to obtain .
We now know and . .
There is a better solution where we find EF directly instead of in parts (use similarity). The strategy is similar.
Solution 4(Coordinate Bash)
To keep things simple, we will use coordinates in only the first quadrant. The picture will look like the diagram above reflected over the -axis.It is also worth noting the will lie on the axis and on the . Let be the origin, , , and . We can express segment as the line . Since is perpendicular to , and we know that lies on it, we can use this information to find that segment is on the line . Since and are on the and axis, respectively, we plug in for and , we find that point is at , and point is at . Applying the distance formula, we obtain that = . Binderclips1 01:04, 10 June 2020 (EDT)
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