## Thursday, January 23, 2014

### Three Pole Problems : Similar triangles

Question: If you know the length of x and y, and the whole length of $$\overline {AB}$$,

A: what is the ratio of a to b and

B: what is the length of z.

Solution for question A:
$$\Delta$$ABC and $$\Delta$$AFE are similar so $$\dfrac {z} {x}=\dfrac {b} {a+b}$$. -- equation 1
Cross multiply and you have z ( a + b ) = bx

$$\Delta$$BAD and $$\Delta$$BFE are similar so $$\dfrac {z} {y}=\dfrac {a} {a+b}$$. -- equation 2
Cross multiply and you have z ( a + b ) = ay

bx = ay so $$\dfrac {x} {y}=\dfrac {a} {b}$$  same ratio

Solution for question B:
Continue with the previous two equations, if you add equation 1 and equation 2, you have:
$$\dfrac {z} {x}+\dfrac {z} {y}=\dfrac {b} {a+b}+\dfrac {a} {a+b}$$
$$\dfrac {zy+zx } {xy}=1$$ $$\rightarrow$$ z = $$\dfrac {xy} {x+y}$$

Applicable question:

$$\overline {CD}=15$$ and you know $$\overline {DB}:\overline {BC}=20:30=2:3$$
so  $$\overline {DB}=6$$ and $$\overline {BC}=9$$

$$\overline {AB}=\dfrac {20\times 30} {\left( 20+30\right) }$$ = 12