$M=\frac{28-1}{6-(-4)}=\frac{7}{12}$
e) Find the perpendicular bisector of cansains $P \& Q$.
$\begin{array}{l}
y=\frac{-12}{7} \times\left(+\frac{12}{7}+\frac{9}{2}\right) \\
\quad \frac{48}{28}+\frac{126}{28}=\frac{178}{28}=\frac{87}{14} \\
y=\frac{-12}{7} \times \frac{87}{14}
\end{array}$
Answer (1)
Correct the initial slope calculation using the coordinates of points P and Q .
Calculate the midpoint M of the line segment PQ using the midpoint formula.
Determine the slope of the perpendicular bisector m ⊥ as the negative reciprocal of the slope of PQ .
Use the point-slope form to derive the equation of the perpendicular bisector and simplify to slope-intercept form: y = − 27 10 x + 54 803 .
Explanation
Problem Analysis and Setup Let's break down how to find the perpendicular bisector of the line segment PQ . First, we need to identify the coordinates of points P and Q and correct the initial slope calculation.
Correcting the Slope and Identifying Points From the given slope calculation M = 6 − ( − 4 ) 28 − 1 = 12 7 , we can infer the coordinates of points P and Q . However, the slope calculation is incorrect. The correct slope calculation is: M = 6 − ( − 4 ) 28 − 1 = 10 27 So, we have P = ( − 4 , 1 ) and Q = ( 6 , 28 ) .
Calculating the Midpoint Next, we calculate the midpoint M of the line segment PQ . The midpoint is given by: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) = ( 2 − 4 + 6 , 2 1 + 28 ) = ( 2 2 , 2 29 ) = ( 1 , 2 29 ) So, the midpoint M is ( 1 , 2 29 ) .
Calculating the Slope of the Perpendicular Bisector Now, we calculate the slope of the line segment PQ using the corrected slope: m PQ = x 2 − x 1 y 2 − y 1 = 6 − ( − 4 ) 28 − 1 = 10 27 The slope of the perpendicular bisector m ⊥ is the negative reciprocal of the slope of PQ :
m ⊥ = − m PQ 1 = − 10 27 1 = − 27 10
Finding the Equation of the Perpendicular Bisector Using the point-slope form of a line, we find the equation of the perpendicular bisector: y − y M = m ⊥ ( x − x M ) y − 2 29 = − 27 10 ( x − 1 ) Now, we simplify the equation to the slope-intercept form y = m x + b :
y = − 27 10 x + 27 10 + 2 29 y = − 27 10 x + 54 20 + 54 783 y = − 27 10 x + 54 803
Final Answer Therefore, the equation of the perpendicular bisector of the line segment PQ is: y = − 27 10 x + 54 803
Examples
In architecture, finding the perpendicular bisector is crucial when designing symmetrical structures or dividing spaces equally. For instance, if you have two key points in a building plan and need to create a wall that perfectly bisects the line connecting them at a 90-degree angle, you would use the principles of perpendicular bisectors to ensure symmetry and balance in the design. This ensures that the wall is equidistant from both points, creating a harmonious and structurally sound division of space.
