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
