Points A and B have coordinates of (5, 2) and (10, -1) respectively. What is the equation of the perpendicular bisector of AB.
The Correct Answer and Explanation is:
To find the equation of the perpendicular bisector of the line segment connecting points A(5, 2) and B(10, -1), follow these steps:
Step 1: Find the midpoint of AB
The midpoint of a line segment is found by averaging the x-coordinates and the y-coordinates of the two endpoints. Midpoint=(x1+x22,y1+y22)\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)Midpoint=(2×1+x2,2y1+y2)
For points A(5, 2) and B(10, -1): Midpoint=(5+102,2+(−1)2)=(152,12)\text{Midpoint} = \left( \frac{5 + 10}{2}, \frac{2 + (-1)}{2} \right) = \left( \frac{15}{2}, \frac{1}{2} \right)Midpoint=(25+10,22+(−1))=(215,21)
So, the midpoint is (7.5,0.5)\left( 7.5, 0.5 \right)(7.5,0.5).
Step 2: Find the slope of AB
The slope of a line is given by the formula: slope=y2−y1x2−x1\text{slope} = \frac{y_2 – y_1}{x_2 – x_1}slope=x2−x1y2−y1
For points A(5, 2) and B(10, -1): slope of AB=−1−210−5=−35\text{slope of AB} = \frac{-1 – 2}{10 – 5} = \frac{-3}{5}slope of AB=10−5−1−2=5−3
Step 3: Find the slope of the perpendicular bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of AB. Since the slope of AB is −3/5-3/5−3/5, the slope of the perpendicular bisector will be 5/35/35/3 (because the negative reciprocal of −3/5-3/5−3/5 is 5/35/35/3).
Step 4: Use point-slope form to find the equation of the perpendicular bisector
The point-slope form of a line equation is: y−y1=m(x−x1)y – y_1 = m(x – x_1)y−y1=m(x−x1)
where mmm is the slope and (x1,y1)(x_1, y_1)(x1,y1) is a point on the line. Using the midpoint (7.5,0.5)(7.5, 0.5)(7.5,0.5) and the slope 5/35/35/3, the equation becomes: y−0.5=53(x−7.5)y – 0.5 = \frac{5}{3}(x – 7.5)y−0.5=35(x−7.5)
Step 5: Simplify the equation
First, distribute 53\frac{5}{3}35 on the right side: y−0.5=53x−53×7.5y – 0.5 = \frac{5}{3}x – \frac{5}{3} \times 7.5y−0.5=35x−35×7.5 y−0.5=53x−12.5y – 0.5 = \frac{5}{3}x – 12.5y−0.5=35x−12.5
Next, add 0.5 to both sides to isolate yyy: y=53x−12.5+0.5y = \frac{5}{3}x – 12.5 + 0.5y=35x−12.5+0.5 y=53x−12y = \frac{5}{3}x – 12y=35x−12
Thus, the equation of the perpendicular bisector of AB is: y=53x−12y = \frac{5}{3}x – 12y=35x−12
Conclusion
The equation of the perpendicular bisector of the line segment joining points A(5, 2) and B(10, -1) is y=53x−12y = \frac{5}{3}x – 12y=35x−12. This line passes through the midpoint of the segment and is perpendicular to it.
