4.3 Q-11

Question Statement

Find the equation of the perpendicular bisector joining the points $A(3,5)$ and $B(9,6)$.

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Background and Explanation

To solve this problem, we need to:

1. Find the midpoint of the line segment joining points $A$ and $B$.
2. Determine the slope of the line through $A$ and $B$.
3. Use the fact that the perpendicular bisector of a line segment has a slope that is the negative reciprocal of the original slope.
4. Find the equation of the perpendicular bisector using the point-slope form.

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Solution

Step 1: Find the Midpoint of $\overline{AB}$

The midpoint $M$ of a line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

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

Substituting the coordinates of points $A(3, 5)$ and $B(9, 6)$:

$$M = \left(\frac{3 + 9}{2}, \frac{5 + 6}{2}\right) = (6, \frac{11}{2})$$

So, the midpoint of $\overline{AB}$ is $M(6, \frac{11}{2})$.

Step 2: Find the Slope of Line $\overline{AB}$

The slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Substituting the coordinates of $A(3, 5)$ and $B(9, 6)$:

$$m = \frac{6 - 5}{9 - 3} = \frac{1}{6}$$

So, the slope of the line joining $A$ and $B$ is $\frac{1}{6}$.

Step 3: Find the Slope of the Perpendicular Bisector

The slope of the perpendicular bisector is the negative reciprocal of the slope of line $\overline{AB}$. If the slope of $\overline{AB}$ is $\frac{1}{6}$, the slope of the perpendicular bisector is:

$$m_{\text{perp}} = -6$$

Step 4: Write the Equation of the Perpendicular Bisector

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

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

We know the slope $m = -6$ and the point on the line is the midpoint $(6, \frac{11}{2})$. Substituting these values into the point-slope form:

$$y - \frac{11}{2} = -6(x - 6)$$

Simplify the equation:

$$y - \frac{11}{2} = -6x + 36$$

Multiply through by 2 to eliminate the fraction:

$$2y - 11 = -12x + 72$$

Now, rearrange the terms to put the equation in standard form:

$$12x + 2y - 83 = 0$$

So, the equation of the perpendicular bisector is:

$$\boxed{12x + 2y - 83 = 0}$$

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Key Formulas or Methods Used

1. Midpoint Formula:

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

2. Slope Formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

3. Negative Reciprocal for perpendicular lines: If the slope of one line is $m$, the slope of the perpendicular line is $-\frac{1}{m}$.

4. Point-Slope Form of a Line:

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

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Summary of Steps

1. Find the midpoint of $\overline{AB}$ using the midpoint formula: $M = (6, \frac{11}{2})$.
2. Calculate the slope of line $\overline{AB}$: $m = \frac{1}{6}$.
3. Find the slope of the perpendicular bisector by taking the negative reciprocal: $m_{\text{perp}} = -6$.
4. Use the point-slope form to write the equation of the perpendicular bisector, simplifying it to standard form: $12x + 2y - 83 = 0$.