Question

Jamie wants to determine if the point (5, 6) lies on the perpendicular bisector of the segments with endpoints (2, 3) and (10,5). Explain how she can do this.

Answer 1

To determine if the point (5, 6) lies on the perpendicular bisector of the segments with endpoints (2, 3) and (10, 5), Jamie can follow these steps: Find the midpoint of the line segment, calculate the slope of the segment, find the slope of the perpendicular bisector, determine the equation of the perpendicular bisector, and substitute the coordinates of the point into the equation.

Step-by-step explanation:

To determine if the point (5, 6) lies on the perpendicular bisector of the segments with endpoints (2, 3) and (10,5), Jamie can follow these steps:

1. Find the midpoint of the line segment between the two endpoints using the midpoint formula: (x, y) = ((x1 + x2)/2, (y1 + y2)/2). In this case, the midpoint is ((2 + 10)/2, (3 + 5)/2) = (6, 4).
2. Calculate the slope of the segment connecting the two endpoints using the slope formula: m = (y2 - y1)/(x2 - x1). In this case, the slope is (5 - 3)/(10 - 2) = 2/8 = 1/4.
3. The perpendicular bisector will have a slope that is the negative reciprocal of the slope of the segment: -1/m. In this case, the slope of the perpendicular bisector is -1/(1/4) = -4.
4. Determine the equation of the perpendicular bisector using the slope-intercept form: y = mx + b. Substitute the midpoint coordinates (6, 4) and the perpendicular slope -4 to find the value of b. The equation becomes 4 = -4(6) + b. Solving for b, we get b = 28.
5. Substitute the coordinates of the point (5, 6) into the equation of the perpendicular bisector: 6 = -4(5) + 28. If the equation is satisfied, then the point lies on the perpendicular bisector.

After substituting, we find that 6 = -20 + 28, which is true. Therefore, the point (5, 6) lies on the perpendicular bisector of the segments with endpoints (2, 3) and (10, 5).