Find a general form of an equation for the perpendicular bisector of the segment AB. A(12,-3), B(6,-15)
Answer (2)
Find the midpoint M of the line segment A B using the midpoint formula: M = ( 9 , − 9 ) .
Calculate the slope of the line segment A B using the slope formula: m A B = 2 .
Determine the slope of the perpendicular bisector as the negative reciprocal of m A B : m ⊥ = − 2 1 .
Use the point-slope form to find the equation of the perpendicular bisector and convert it to general form: x + 2 y + 9 = 0 .
Explanation
Problem Analysis We are given two points, A ( 12 , − 3 ) and B ( 6 , − 15 ) , and we want to find the equation of the perpendicular bisector of the line segment A B .
Find the Midpoint First, we need to find the midpoint M of the line segment A B . The midpoint formula is given by: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) Substituting the coordinates of A and B , we get: M = ( 2 12 + 6 , 2 − 3 + ( − 15 ) ) = ( 2 18 , 2 − 18 ) = ( 9 , − 9 ) So, the midpoint M is ( 9 , − 9 ) .
Find the Slope of AB Next, we need to find the slope of the line segment A B . The slope formula is given by: m A B = x 2 − x 1 y 2 − y 1 Substituting the coordinates of A and B , we get: m A B = 6 − 12 − 15 − ( − 3 ) = − 6 − 15 + 3 = − 6 − 12 = 2 So, the slope of A B is 2 .
Find the Slope of the Perpendicular Bisector Now, we need to find the slope of the perpendicular bisector. The slope of the perpendicular bisector is the negative reciprocal of the slope of A B . Therefore, m ⊥ = − m A B 1 = − 2 1 So, the slope of the perpendicular bisector is − 2 1 .
Find the Equation of the Perpendicular Bisector We have the midpoint M ( 9 , − 9 ) and the slope of the perpendicular bisector m ⊥ = − 2 1 . We can use the point-slope form of a line to find the equation of the perpendicular bisector: y − y 1 = m ⊥ ( x − x 1 ) Substituting the values, we get: y − ( − 9 ) = − 2 1 ( x − 9 ) y + 9 = − 2 1 x + 2 9 To convert this to general form, we first multiply by 2 to eliminate the fraction: 2 ( y + 9 ) = 2 ( − 2 1 x + 2 9 ) 2 y + 18 = − x + 9 Now, rearrange the equation to the general form A x + B y + C = 0 :
x + 2 y + 18 − 9 = 0 x + 2 y + 9 = 0 Thus, the general form of the equation for the perpendicular bisector is x + 2 y + 9 = 0 .
Final Answer The equation of the perpendicular bisector of the segment A B is x + 2 y + 9 = 0 .
Examples
In architecture, determining the perpendicular bisector is crucial when designing symmetrical structures or dividing spaces equally. For instance, if two anchor points of a bridge need to be equidistant from a central support, the perpendicular bisector helps locate the support's position accurately. This ensures balanced weight distribution and structural integrity, showcasing a practical application of geometric principles in engineering.
The equation of the perpendicular bisector of the segment AB with points A(12,-3) and B(6,-15) is determined by first finding the midpoint and the slope of the segment. The final equation in general form is x + 2 y + 9 = 0 .
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