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.