Question
B is the midpoint of AC and E is the midpoint of BD. If A(-9,-4), C(-1,6), and E(-4,-3), find the coordinates of D.
Ask by Watson Frank.
Mar 28,2025 18:38
UpStudy AI Solution
Tutor-Verified Answer
Answer
The coordinates of \( D \) are \( (-3, -7) \).
Solution
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Mind Expander
To find the coordinates of point \( D \), we first need to determine the coordinates of point \( B \), which is the midpoint of \( \overline{A C} \).
The coordinates of \( B \) can be calculated using the midpoint formula:
\[
B = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-9 + (-1)}{2}, \frac{-4 + 6}{2} \right) = \left( \frac{-10}{2}, \frac{2}{2} \right) = (-5, 1)
\]
Now, we know that \( E \) is the midpoint of \( \overline{B D} \). We can apply the midpoint formula again for points \( B \) and \( D \):
\[
E = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right)
\]
Substituting in the coordinates of \( E \) and \( B \):
\[
(-4, -3) = \left( \frac{-5 + x_D}{2}, \frac{1 + y_D}{2} \right)
\]
Now, we can set up the equations to solve for \( x_D \) and \( y_D \):
1. For the x-coordinates:
\[
-4 = \frac{-5 + x_D}{2}
\]
Multiplying both sides by 2:
\[
-8 = -5 + x_D \implies x_D = -8 + 5 = -3
\]
2. For the y-coordinates:
\[
-3 = \frac{1 + y_D}{2}
\]
Multiplying both sides by 2:
\[
-6 = 1 + y_D \implies y_D = -6 - 1 = -7
\]
Thus, the coordinates of point \( D \) are \( (-3, -7) \).
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