Problem Breakdown:

We are given three points:
$A(-3, -4)$, $B(1, 2)$, and $C(4, -3)$.

1. Determine the equation of the height drawn from $C$:
   The height from $C$ in $\triangle ABC$ is a line that passes through $C$ and is perpendicular to the side $AB$.

2. Determine the equation of the perpendicular bisector of segment $[AB]$:
   The perpendicular bisector is a line that is perpendicular to $AB$ and passes through its midpoint.

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1. Height from $C$:

Step 1. Find the slope of $AB$:

The slope formula is:
$m = \frac{y_2 - y_1}{x_2 - x_1}.$
Substituting $A(-3, -4)$ and $B(1, 2)$:
$m_{AB} = \frac{2 - (-4)}{1 - (-3)} = \frac{6}{4} = \frac{3}{2}.$

Step 2. Find the slope of the height:

The height from $C$ is perpendicular to $AB$, so its slope is the negative reciprocal of $m_{AB}$:
$m_{\text{height}} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{3}{2}} = -\frac{2}{3}.$

Step 3. Write the equation of the height:

The equation of a line is given by:
$y - y_1 = m(x - x_1),$
where $(x_1, y_1) = (4, -3)$ and $m = -\frac{2}{3}$:
$y - (-3) = -\frac{2}{3}(x - 4).$
Simplify:
$y + 3 = -\frac{2}{3}x + \frac{8}{3}.$
$y = -\frac{2}{3}x + \frac{8}{3} - 3.$
$y = -\frac{2}{3}x - \frac{1}{3}.$

The equation of the height is:
$y = -\frac{2}{3}x - \frac{1}{3}.$

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2. Perpendicular Bisector of $[AB]$:

Step 1. Find the midpoint of $AB$:

The midpoint formula is:
$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right).$
Substituting $A(-3, -4)$ and $B(1, 2)$:
$M = \left( \frac{-3 + 1}{2}, \frac{-4 + 2}{2} \right) = \left( -1, -1 \right).$

Step 2. Find the slope of the perpendicular bisector:

The slope of $AB$ is $\frac{3}{2}$, so the slope of the perpendicular bisector is the negative reciprocal:
$m_{\text{bisector}} = -\frac{1}{m_{AB}} = -\frac{2}{3}.$

Step 3. Write the equation of the perpendicular bisector:

Using the point-slope form $y - y_1 = m(x - x_1)$, where $(x_1, y_1) = (-1, -1)$ and $m = -\frac{2}{3}$:
$y - (-1) = -\frac{2}{3}(x - (-1)).$
Simplify:
$y + 1 = -\frac{2}{3}(x + 1).$
$y + 1 = -\frac{2}{3}x - \frac{2}{3}.$
$y = -\frac{2}{3}x - \frac{2}{3} - 1.$
$y = -\frac{2}{3}x - \frac{5}{3}.$

The equation of the perpendicular bisector is:
$y = -\frac{2}{3}x - \frac{5}{3}.$

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Summary of Results:

1. Equation of the height from $C$:
   $y = -\frac{2}{3}x - \frac{1}{3}.$
2. Equation of the perpendicular bisector of $[AB]$:
   $y = -\frac{2}{3}x - \frac{5}{3}.$

Do you want me to elaborate further on any step or concept? 😊

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Related Questions:

1. What is the equation of the side $AB$ in the triangle?
2. How do you find the coordinates of the centroid of $\triangle ABC$?
3. What are the slopes of the other two heights in $\triangle ABC$?
4. Can you calculate the area of $\triangle ABC$?
5. How do you find the circumcenter of $\triangle ABC$?

Tip:

Always verify perpendicularity by checking if the product of two slopes is $-1$.