Question

identify the equation in point-slope form for the perpendicular bisector of the segment with endpoints e(−7,6) and g(9,−2).

Answer 1

The equation in point-slope form for the perpendicular bisector of the segment with endpoints (-7,6) and (9,-2) is y = 2x - 2.

Step-by-step explanation:

The equation of the perpendicular bisector of a line segment is in the form y = mx + b, where m is the slope of the segment and b is the y-intercept. To find the slope of the segment, we use the slope formula: m = (y2 - y1) / (x2 - x1). Plugging in the coordinates of the endpoints e(-7,6) and g(9,-2), we find the slope of the segment to be m = (-2 - 6) / (9 - (-7)) = -8 / 16 = -1/2.

The slope of the perpendicular bisector will be the negative reciprocal of the slope of the segment, so the slope of the perpendicular bisector is 2. To find the midpoint of the segment, we use the midpoint formula: (x, y) = ((x1 + x2) / 2, (y1 + y2) / 2). Plugging in the coordinates of the endpoints, we find the midpoint to be ((-7 + 9) / 2, (6 + (-2)) / 2) = (1, 2).

Now that we have the slope and a point on the perpendicular bisector, we can use the point-slope form of the equation: y - y1 = m(x - x1). Substituting in the values, we get y - 2 = 2(x - 1), which simplifies to y = 2x - 2.

Answer 2

The equation in point-slope form for the perpendicular bisector of the segment with endpoints e(−7,6) and g(9,−2) is:

`y-2=2(x-1)`

The perpendicular bisector of a line segment is a line that passes through the midpoint of the segment and is perpendicular to the line segment.

The formula for the midpoint of a line segment with endpoints
`\left(x_1, y_1\right) \text { and }\left(x_2, y_2\right)` is given by:

`\left((x_1+x_2)/(2), (y_1+y_2)/(2)\right)`

For the given endpoints e(−7,6) and g(9,−2), the midpoint is:

`\left((-7+9)/(2), (6+(-2))/(2)\right)=(1,2)`

Now, the slope of the line segment between e(−7,6) and g(9,−2) is given by the formula:

`m=(y_2-y_1)/(x_2-x_1)`

`m=(-2-6)/(9-(-7))=(-8)/(16)=-(1)/(2)`

The negative reciprocal of -1/2 is 2, which is the slope of the perpendicular bisector. Now, we can use the point-slope form of a line:

`y-y_1=m\left(x-x_1\right)`

Substitute the midpoint (1,2) and the slope m=2:

`y-2=2(x-1)`

This is the required equation in point-slope form for the given endpoints of perpendicular bisector.