In △ ABC, AB=5, BC=7, AC=9 and D is on AC with BD=5. Find the ratio AD:DC.

1989 AMC 12 Problem 15

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15.

\triangle ABC,

AB=5,

BC=7,

AC=9

and

D

is on

\overline{AC}

with

BD=5.

Find the ratio

AD:DC.

4:3

7:5

11:6

13:5

19:8

Let

AD=m

and

DC=9-m

Apply Stewart’s Theorem to cevian

BD

Let

AD=m

and

DC=9-m.

Stewart’s Theorem gives

\begin{gathered} 7^2m+5^2(9-m) \\ =9\bigl(5^2+m(9-m)\bigr). \end{gathered}

Simplifying yields

9m^2-57m=0,

m=\frac{19}{3}.

Then

DC=\frac{8}{3},

and

AD:DC=19:8.

Thus the correct answer is E.