To find the coordinates of point $D$, which is the midpoint of the hypotenuse of $\triangle ABC$, we need to identify the two endpoints of the hypotenuse and then use the midpoint formula.

Hypotenuse of $\triangle ABC$

In a right triangle, the hypotenuse is the side opposite the right angle. Based on the diagram, it seems the hypotenuse is the line connecting points $A(3, 6)$ and $B(7, -3)$.

Midpoint Formula

The midpoint $D(x, y)$ of a line segment connecting two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$D\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Calculation

Let’s calculate the coordinates of point $D$ using the points $A(3, 6)$ and $B(7, -3)$.

$D_x = \frac{3 + 7}{2} = \frac{10}{2} = 5$ $D_y = \frac{6 + (-3)}{2} = \frac{6 - 3}{2} = \frac{3}{2} = 1.5$

Thus, the coordinates of point $D$ are $(5, 1.5)$.

Would you like to proceed with the next question or delve deeper into this result?

Here are 5 related questions to explore:

1. How do you verify that line $AB$ is indeed the hypotenuse?
2. What is the length of the hypotenuse $AB$?
3. How do you find the equation of the line passing through point $D$?
4. Can you calculate the slope of line $AB$?
5. How does the midpoint relate to the circumcenter of a right triangle?

Tip: The midpoint of the hypotenuse in a right triangle is also the circumcenter, meaning it's equidistant from all three vertices.