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We are tasked with finding the coordinates of midpoint E in the triangle, given the points \( A(-3a, b) \), \( C(-a, -5b) \), and the midpoint formula:

\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Midpoint E is between points \( A(-3a, b) \) and \( C(-a, -5b) \). Using the midpoint formula, let’s calculate the coordinates of E step by step.

1. **X-coordinate of E:**
   Using \( x_1 = -3a \) and \( x_2 = -a \),
   \[
   x_E = \frac{-3a + (-a)}{2} = \frac{-4a}{2} = -2a
   \]

2. **Y-coordinate of E:**
   Using \( y_1 = b \) and \( y_2 = -5b \),
   \[
   y_E = \frac{b + (-5b)}{2} = \frac{-4b}{2} = -2b
   \]

Thus, the coordinates of midpoint E are:
\[
E(-2a, -2b)
\]

Would you like a more detailed explanation, or have any further questions?

Here are 5 related questions you might find interesting:
1. How would the midpoint formula change if we used 3D coordinates?
2. How can we verify that E is indeed the midpoint of A and C?
3. What is the distance between points A and C using the distance formula?
4. How can the midpoint concept be applied in geometry for partitioning line segments?
5. What would happen if one of the points had a negative value for \( a \) or \( b \)?

**Tip:** The midpoint formula is a simple average of the coordinates of two points, making it highly useful in various geometric calculations!

Find the coordinates of midpoint E using the given points A(-3a, b) and C(-a, -5b).

Learn how to find the coordinates of midpoint E using the midpoint formula, given points A(-3a, b) and C(-a, -5b). This problem explores key geometric concepts and demonstrates how to apply the midpoint formula step-by-step. Suitable for students in Grades 8-10.

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