The midpoint of the linesegment joiningA(2a,4)andB(-2,3b)isM(1,2a+1)

Given the problem, we're dealing with finding the relationship between the coordinates of the points \( A(2a, 4) \), \( B(-2, 3b) \), and the midpoint \( M(1, 2a + 1) \) of the line segment joining \( A \) and \( B \).

### Formula for the Midpoint:
The midpoint \( M(x_m, y_m) \) of a line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:

\[
x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}
\]

### Applying the Formula:
For point \( A(2a, 4) \) and point \( B(-2, 3b) \), the midpoint \( M(1, 2a + 1) \) gives us the following equations:

1. **For the \( x \)-coordinate:**

\[
\frac{2a + (-2)}{2} = 1
\]

Simplifying:

\[
\frac{2a - 2}{2} = 1
\]

\[
2a - 2 = 2
\]

\[
2a = 4
\]

\[
a = 2
\]

2. **For the \( y \)-coordinate:**

\[
\frac{4 + 3b}{2} = 2a + 1
\]

Since \( a = 2 \), substitute \( a = 2 \) into the equation:

\[
\frac{4 + 3b}{2} = 2(2) + 1
\]

\[
\frac{4 + 3b}{2} = 5
\]

Multiply both sides by 2:

\[
4 + 3b = 10
\]

\[
3b = 6
\]

\[
b = 2
\]

### Final Answer:
The values of \( a \) and \( b \) are both \( 2 \).

Would you like more details or have any further questions?

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Here are 5 related questions to consider:
1. How do you find the distance between two points \( A \) and \( B \)?
2. What is the formula for the slope of the line through points \( A \) and \( B \)?
3. How do you derive the equation of the line passing through two points?
4. What are the coordinates of the centroid of a triangle with vertices \( A \), \( B \), and \( C \)?
5. Can you find the equation of a perpendicular bisector of a line segment?

**Tip:** Always double-check your calculations by substituting the found values back into the original equations to ensure accuracy.

Discover how to find the midpoint of a line segment problem solution step-by-step. This problem involves coordinate geometry and applies the midpoint formula to find the coordinates of points A(2a, 4) and B(-2, 3b), given midpoint M(1, 2a + 1). Suitable for Grades 10-12 students.

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