2003 AMC 10A Problems/Problem 22
In rectangle , we have , , is on with , is on with , line intersects line at , and is on line with . Find the length of .
(Vertical angles are equal).
(Both are 90 degrees).
(Alt. Interior Angles are congruent).
Therefore and are similar. and are also similar.
is 9, therefore must equal 5. Similarly, must equal 3.
Because and are similar, the ratio of and , must also hold true for and . , so is of . By Pythagorean theorem, .
Since is a rectangle, .
Since is a rectangle and , .
Since is a rectangle, .
So, is a transversal, and .
This is sufficient to prove that and .
Since can't have 2 different lengths, both expressions for must be equal.
Solution 3 (fastest)
We extend such that it intersects at . Since is a rectangle, it follows that , therefore, . Let . From the similarity of triangles and , we have the ratio (as , and ). and are the altitudes of and , respectively. Thus, , from which we have , thus
Since and we have Thus, Suppose and Thus, we have Additionally, now note that which is pretty obvious from insight, but can be proven by AA with extending to meet From this new pair of similar triangles, we have Therefore, we have by combining those two equations, Solving, we have and therefore
Since there are only lines, you can resort to coordinate bashing. Let . Three lines, line , line , and line , intersect at . Our goal is to find the y-coordinate of that intersection point.
Line passes through and . Therefore the slope is and the line is which is
Line passes through and . Therefore the slope is and the line is which simplifies to
We solve the system of equations with these three lines. First we plug in
Next, we solve for k. Therefore . The y-coordinate of this intersection point is indeed our answer. ~superagh
Solution 6 (simple coordinates)
Let be the origin of our coordinate system. Now line has equation . We can use point-slope form to find the equation for line . First, we know that its slope is , and we know that it passes through , so line has equation . Solving for the intersection by letting , we get . Plugging this into our equation for line gives us , so ~chrisdiamond10
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