Let's break down and solve the problem step by step.

We are given that the line $L_1$ has the equation:

$3x + 4y = 24$

This line cuts the $x$-axis at point $A$ and the $y$-axis at point $B$.

Part (a): Coordinates of A and B

To find the coordinates of $A$ (on the $x$-axis) and $B$ (on the $y$-axis):

• At $A$, the line intersects the $x$-axis, so $y = 0$: $3x + 4(0) = 24 \implies 3x = 24 \implies x = 8$ Hence, the coordinates of $A$ are $(8, 0)$.

• At $B$, the line intersects the $y$-axis, so $x = 0$: $3(0) + 4y = 24 \implies 4y = 24 \implies y = 6$ Hence, the coordinates of $B$ are $(0, 6)$.

Part (b): Coordinates of M

$M$ is the midpoint of line segment $[AB]$. The midpoint formula is given by:

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

Substituting the coordinates of $A (8, 0)$ and $B (0, 6)$:

$M = \left( \frac{8 + 0}{2}, \frac{0 + 6}{2} \right) = (4, 3)$

Thus, the coordinates of $M$ are $(4, 3)$.

Part (c): Equation of Line $L_2$

We are given that line $L_2$ passes through points $M(4, 3)$ and $C(0, -2)$. To find the equation of line $L_2$, we first find the slope using the slope formula:

$\text{slope of } L_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - (-2)}{4 - 0} = \frac{5}{4}$

Now that we have the slope, we use the point-slope form of a line equation:

$y - y_1 = m(x - x_1)$

Using point $M(4, 3)$:

$y - 3 = \frac{5}{4}(x - 4)$

Simplifying:

$y - 3 = \frac{5}{4}x - 5 \implies y = \frac{5}{4}x - 5 + 3 \implies y = \frac{5}{4}x - 2$

Thus, the equation of line $L_2$ is:

$y = \frac{5}{4}x - 2$

Part (d): Lengths of MC and AC

1. Length of $MC$:

Using the distance formula between points $M(4, 3)$ and $C(0, -2)$:

$MC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(4 - 0)^2 + (3 - (-2))^2} = \sqrt{4^2 + (3 + 2)^2} = \sqrt{16 + 25} = \sqrt{41}$

Thus, $MC = \sqrt{41}$.

2. Length of $AC$:

Using the distance formula between points $A(8, 0)$ and $C(0, -2)$:

$AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(8 - 0)^2 + (0 - (-2))^2} = \sqrt{8^2 + 2^2} = \sqrt{64 + 4} = \sqrt{68}$

Thus, $AC = \sqrt{68}$.

Do you need further clarification on any part of the solution? Here are some related questions you can explore:

1. How do we verify the equation of a line given two points?
2. What are the coordinates of the midpoint of a segment in general?
3. How do you derive the distance formula from the Pythagorean theorem?
4. Can a line be parallel to the x- or y-axis? If so, how would the equation look?
5. How does changing one point on a line affect its slope?

Tip: The distance formula is derived from the Pythagorean theorem and can be used to find the straight-line distance between any two points in a plane.