Find the equation of Line 4 & Line 5.
Understand the Problem
The question is asking to find the equations of two lines: Line 4, which is perpendicular to the line segment connecting points C and D and passes through its midpoint, and Line 5, which is parallel to Line 4 and passes through the point (-3, -4). We will need to calculate the midpoint of CD, determine the slope of CD to find the slope of Line 4 (which is its negative reciprocal), and then formulate the equations for both lines.
Answer
Line 4: $y = -4x + 12$; Line 5: $y = -4x - 16$
Answer for screen readers
The equations of the lines are:
Line 4: $y = -4x + 12$
Line 5: $y = -4x - 16$
Steps to Solve
- Calculate the Midpoint of Segment CD
To find the midpoint, use the midpoint formula:
$$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$
Substituting points C and D where $C(-4, -6)$ and $D(12, -2)$:
$$ M = \left( \frac{-4 + 12}{2}, \frac{-6 - 2}{2} \right) = \left( \frac{8}{2}, \frac{-8}{2} \right) = (4, -4) $$
- Determine the Slope of Line Segment CD
The slope of a line through two points is calculated as:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Using points C and D:
$$ m_{CD} = \frac{-2 - (-6)}{12 - (-4)} = \frac{4}{16} = \frac{1}{4} $$
- Find the Slope of Line 4 (Perpendicular to CD)
For a line perpendicular to another, the slope is the negative reciprocal:
$$ m_{Line4} = -\frac{1}{m_{CD}} = -\frac{1}{\frac{1}{4}} = -4 $$
- Write the Equation of Line 4 Using Point-Slope Form
Point-slope form is given by:
$$ y - y_1 = m(x - x_1) $$
Using point $(4, -4)$ and slope $-4$:
$$ y - (-4) = -4(x - 4) $$
This simplifies to:
$$ y + 4 = -4x + 16 $$
So,
$$ y = -4x + 12 $$
- Write the Equation of Line 5 (Parallel to Line 4)
Line 5 will have the same slope as Line 4, which is $-4$. Using point (-3, -4):
$$ y - y_1 = m(x - x_1) $$
Substituting:
$$ y - (-4) = -4(x - (-3)) $$
This simplifies to:
$$ y + 4 = -4(x + 3) $$
So,
$$ y + 4 = -4x - 12 $$
Thus,
$$ y = -4x - 16 $$
The equations of the lines are:
Line 4: $y = -4x + 12$
Line 5: $y = -4x - 16$
More Information
Line 4 is perpendicular to line segment CD and intersects it at its midpoint, while Line 5 runs parallel to Line 4 and passes through a specified point. This exercise illustrates how to derive equations of lines based on given points and relationships.
Tips
- Failing to accurately calculate the midpoint.
- Incorrectly finding the negative reciprocal for the perpendicular slope.
- Confusing the point-slope form when substituting values.
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