The task is to find the equation of the median $BB_1$ in triangle $\triangle ABC$, where:

• $A(7, 7)$,
• $B(-3, 4)$,
• $C(-8, -7)$.

Solution Steps:

1. Find the midpoint of side $AC$:
   The median $BB_1$ starts at $B$ and passes through the midpoint $B_1$ of $AC$.
   Using the midpoint formula:
   $B_1 = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ Substituting $A(7, 7)$ and $C(-8, -7)$:
   $B_1 = \left(\frac{7 + (-8)}{2}, \frac{7 + (-7)}{2}\right) = \left(\frac{-1}{2}, 0\right)$

2. Find the slope of $BB_1$:
   The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:
   $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting $B(-3, 4)$ and $B_1\left(-\frac{1}{2}, 0\right)$:
   $m = \frac{0 - 4}{-\frac{1}{2} - (-3)} = \frac{-4}{-\frac{1}{2} + 3} = \frac{-4}{\frac{5}{2}} = -\frac{8}{5}$

3. Write the equation of the median $BB_1$:
   The equation of a line is:
   $y - y_1 = m(x - x_1)$ Substituting $m = -\frac{8}{5}$, $x_1 = -3$, and $y_1 = 4$:
   $y - 4 = -\frac{8}{5}(x + 3)$ Simplify:
   $y - 4 = -\frac{8}{5}x - \frac{24}{5}$ $y = -\frac{8}{5}x - \frac{24}{5} + 4$ Convert $4$ to a fraction: $4 = \frac{20}{5}$, so:
   $y = -\frac{8}{5}x - \frac{24}{5} + \frac{20}{5}$ $y = -\frac{8}{5}x - \frac{4}{5}$

Final Answer:

The equation of the median $BB_1$ is:
$y = -\frac{8}{5}x - \frac{4}{5}$

Let me know if you'd like more details or have questions!

Related questions to consider:

1. How do you find the equation of a perpendicular bisector for a triangle's side?
2. Can you derive the equations for all three medians of $\triangle ABC$?
3. What is the centroid of $\triangle ABC$, and how is it calculated?
4. How do you determine if a given point lies on a median?
5. What is the significance of medians in triangle geometry?

Tip: Always double-check your slope calculations to avoid sign errors!