To determine the coordinates of point D that prove AD is a median of triangle ABC, you need to understand the definition of a median in a triangle. A median is a line segment joining a vertex to the midpoint of the opposite side.
For triangle ABC, where points A = (-7, -8), B = (-5, -6), and C = (7, 2), we need to find the midpoint of side BC. To do this, we calculate the midpoint by averaging the coordinates of B and C.
The formula for the midpoint (M) of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Let's apply this formula to points B(-5, -6) and C(7, 2).
1. Compute the x-coordinate of the midpoint:
[tex]\[ \text{midpoint}_x = \frac{-5 + 7}{2} = \frac{2}{2} = 1 \][/tex]
2. Compute the y-coordinate of the midpoint:
[tex]\[ \text{midpoint}_y = \frac{-6 + 2}{2} = \frac{-4}{2} = -2 \][/tex]
Thus, the coordinates of the midpoint D are:
[tex]\[ D = (1, -2) \][/tex]
However, the given answer choices are:
- D(2,4)
- D(1.5)
- D(6.5)
- D(1.4)
It appears that the given choices do not include the correct computed midpoint. There may be a mistake in the provided coordinates or choices.
Rechecking the coordinates of B and C verifies that the computed D(1, -2) is correct using the provided coordinates:
B = (-5, -6) and C = (7, 2).
Since none of the provided answer choices match the computed midpoint D, it's possible the answer set is incorrect. In any event, we've shown from the calculations that the correct midpoint, and therefore the coordinates of point D that would prove AD is a median of triangle ABC, is (1, -2).
