I’m having problems with finding a simple solution to this problem. By making lots of new lines and angles, I can get the equation x + 4/3x = 12. Which gives me the correct answer. But this is supposed to be a first year high school problem, so I am wondering if there is a simpler solution I have overlooked.
I got the same answer you did, but I didn’t have to draw any new lines or angles. You can use similar triangles. Call the top two points M and N, and the bottom two points P and Q. Call the center top intersection T and the bottom intersection B. Finally, call the bottom line segments a and b. Got all that? OK, let’s go.
Triangle MQP is similar to triangle TQB because it shares three angles (angle Q, a right angle, and since X is parallel to line segment MP, it’s “cutting the transversal” MQ, the right central angle is equal to M). In the same manner, PNQ is similar to PTB.
So NQ/(a+b) = TB/a, which means 9/(a+b) = x/a. Similarly, 12/(a+b) = x/b. Multiply the first equation by 4/3 and you get 3a = 4b, which can be written as b = 3a/4
Now sub that back into 9/(a+b) = x/a
9/(a+.75a) = x/a <--------- I just wrote 3/4 as .75 for ease of internet writing.
9/1.75a = x/a
9/1.75 = x
x=36/7.
Here’s a reasonably (though not stunningly) simple way to prove it:
The vertical line labeled X divides the horizontal “base” line into two parts. Call these A and B (A is the longer of the two).
We can express X in two ways:
X / 9 = A / (A + B) which gives X = 9 * A / (A + B)
X / 12 = B / (A + B) which gives X = 12 * B / (A + B )
If we set the expressions of X equal to each other, it’s easy to show that A / B = 12 / 9 (which seemed likely just from looking at the diagram).
From this it quickly follows that X = A * B / (A + B) so X = 36 /7 or 5.14
ETA: Scooped by Chessic Sense