Question

Let the straight line BN which is the reflection of median BM with respect to the bisector of the interior angle B of a triangle ABC divide the side AB in the ratio AN : NC = x : y. If AB = 8 and BC = 12 find the value of $\frac{2x + 3y}{y - x}$

Answer 1

7

Answer 2

We note that the reflection of the median BM about the internal bisector of ∠B gives a line BN that is isogonal to BM. For two isogonal lines from vertex B meeting side AC at M and N respectively, the following relation holds:

$\frac{AN}{NC} = \frac{AB^2}{BC^2}\cdot \frac{AM}{MC}$

Since BM is the median, M divides AC equally, so $AM = MC$ and

$\frac{AN}{NC} = \frac{AB^2}{BC^2}$

Given $AB = 8$ and $BC = 12$, we have

$\frac{AN}{NC} = \frac{8^2}{12^2} = \frac{64}{144} = \frac{4}{9}$.

Thus, taking $AN:NC = x:y = 4:9$, we need to compute:

$\frac{2x + 3y}{y - x} = \frac{2(4) + 3(9)}{9 - 4} = \frac{8 + 27}{5} = \frac{35}{5} = 7$.