In triangle ABC, AD is the median to side BC and M is the midpoint of AD. If vector BM = λ·vector BA + μ·vector BC, find the ordered pair (λ, μ).

Step-by-step solution

1. 1

   Switch to position vectors so the midpoints become averages. Write $\vec A,\vec B,\vec C,\vec D,\vec M$ for the position vectors of the points. The advantage is that a midpoint is just the mean of its endpoints, which turns the geometry into arithmetic on coefficients.

2. 2

   Locate D, then M. $AD$ is the median to $BC$, so $D$ is the midpoint of $BC$:

   $\vec D=\frac{\vec B+\vec C}{2}$

   and $M$ is the midpoint of $AD$:

   $\vec M=\frac{\vec A+\vec D}{2}=\frac{1}{2}\vec A+\frac{1}{4}\vec B+\frac{1}{4}\vec C$

3. 3

   Form the vector BM. Subtract the position vector of $B$:

   $\overrightarrow{BM}=\vec M-\vec B=\frac{1}{2}\vec A+\frac{1}{4}\vec B+\frac{1}{4}\vec C-\vec B=\frac{1}{2}\vec A-\frac{3}{4}\vec B+\frac{1}{4}\vec C$

4. 4

   Write the target combination in the same coordinates. Since $\overrightarrow{BA}=\vec A-\vec B$ and $\overrightarrow{BC}=\vec C-\vec B$,

   $\lambda\overrightarrow{BA}+\mu\overrightarrow{BC}=\lambda\vec A+\mu\vec C-(\lambda+\mu)\vec B$

5. 5

   Match coefficients. Comparing the $\vec A$ terms gives $\lambda=\tfrac12$; comparing the $\vec C$ terms gives $\mu=\tfrac14$.

   $\boxed{(\lambda,\mu)=\left(\tfrac12,\ \tfrac14\right)}$

6. 6

   Verify with the third coefficient. The $\vec B$ term requires $-(\lambda+\mu)=-\tfrac34$, and indeed $\tfrac12+\tfrac14=\tfrac34$. This consistency check matters: the three coefficients of $\vec A,\vec B,\vec C$ in any such expression must sum to $0$, since $\overrightarrow{BM}$ is a difference of points and does not depend on the origin.