# Geometry problem

**URL:** <https://boards.straightdope.com/t/geometry-problem/605368>\
**Category:** Factual Questions\
**Created:** [December 7, 2011, 1:43pm UTC](https://boards.straightdope.com/t/geometry-problem/605368 "2011-12-07T13:43:34Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![Danger\_Man](https://avatars.discourse-cdn.com/v4/letter/d/6de8d8/32.png) [@Danger\_Man](https://boards.straightdope.com/u/Danger_Man)\
**Post date:** [December 7, 2011, 1:43pm UTC](https://boards.straightdope.com/t/geometry-problem/605368/1 "2011-12-07T13:43:34Z")

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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.

[http://dl.dropbox.com/u/81226/oppgave.png](http://dl.dropbox.com/u/81226/oppgave.png)

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**Author:** ![Chessic\_Sense](https://avatars.discourse-cdn.com/v4/letter/c/7c8e57/32.png) [@Chessic\_Sense](https://boards.straightdope.com/u/Chessic_Sense)\
**Post date:** [December 7, 2011, 2:39pm UTC](https://boards.straightdope.com/t/geometry-problem/605368/2 "2011-12-07T14:39:36Z")

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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.

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**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\
**Post date:** [December 7, 2011, 2:49pm UTC](https://boards.straightdope.com/t/geometry-problem/605368/3 "2011-12-07T14:49:16Z")

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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