On graph paper, draw $ΔABC$ if $A(2, 4)$, $B(9, 5)$, and $C(4, 10)$.   

1. Verify that $D(3, 7)$ is a midpoint of $\overline{AC}$.

   Step (a):

   To find the midpoint between two points (average $x$-coordinates, average $y$-coordinates).

2. Write the equation of the line through points $D$ and $B$.

   Hint (b):

   Read Math Notes box in Lesson 2.3.2 on linear equations from slopes and/or points.

   Step 1(b):

   Calculate the slope between points $D$ and $B$.

   Step 2(b):

   Using the slope, the $x$- and $y$-coordinates from point $D$ or $B$, and slope intercept form ($y = mx + b$). Solve for $b$.

3. Is $\overline{BD}$ a height of $ΔABC$? Use slope to show that $\overline{BD}$ is perpendicular to $\overline{AC}$.

   Hint (c):

   Compare the slopes of segment $\overline{BD}$ and segment $\overline{AC}$. What do you notice?

4. What is the perimeter and area of $ΔABC$?

   Step (d):

   Use the Pythagorean Theorem to find the length of all missing sides.

   Answer (d):

   Perimeter $≈ 20.467$ units, Area $= 20\text{ units}^2$