A theorem from elementary geometry states: If P₁ and P₂ are on opposite sides of a line segment and an equal distance from the endpoints, then the line through P₁ and P₂ is a perpendicular bisector of the segment. Verify this is true for the segment and points shown.

Question

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The Answer is First show that the conditions of the statement are met.
Although it is evident from the graph that P₁ and P₂ are on opposite sides of the line segment

First show that the conditions of the statement are met. Although it is evident from the graph that P<sub>1</sub> and P<sub>2</sub> are on opposite sides of the line segment , it can be proven using the equation of the line . Find the slope of this line using the endpoints of , R (-6, 7) and S (2, -5) and the slope formula: and use the slope-point to find the equation of the line: It can then be shown that P<sub>1</sub>(1, 3) is in the half-plane : and P<sub>2</sub>(-8, -3) is in the half-plane : and they are therefore on opposite sides of the line segment . Next, use the distance formula to show that P<sub>1</sub>(1, 3) is an equal distance from the endpoints R (-6, 7) and S (2, -5): d(P<sub>1</sub>R) d(P<sub>1</sub>S) and that P<sub>2</sub>(-8, -3) is an equal distance from the endpoints R (-6, 7) and S (2, -5): d(P<sub>2</sub>R) d(P<sub>2</sub>S) Now show that the line is a perpendicular bisector of . Use the slope formula to find the slope of the line through P<sub>1</sub>(1, 3) and P<sub>2</sub>(-8, -3):<sub> </sub> <sub> </sub> Since from first step, and are perpendicular. Use the point-slope formula to find the equation of : Find the midpoint of : Show is on : Therefore is a perpendicular bisector of .

, it can be proven using the equation of the line

.
Find the slope of this line using the endpoints of

, R (-6, 7) and S (2, -5) and the slope formula:

and use the slope-point to find the equation of the line:

It can then be shown that P₁(1, 3) is in the half-plane

:

and P₂(-8, -3) is in the half-plane

:

and they are therefore on opposite sides of the line segment

.
Next, use the distance formula to show that P₁(1, 3) is an equal distance from the endpoints R (-6, 7) and S (2, -5):
d(P₁R)

d(P₁S)

and that P₂(-8, -3) is an equal distance from the endpoints R (-6, 7) and S (2, -5):
d(P₂R)

d(P₂S)

Now show that the line

is a perpendicular bisector of

.
Use the slope formula to find the slope of the line through P₁(1, 3) and P₂(-8, -3):

Since

from first step,

and

are perpendicular.
Use the point-slope formula to find the equation of

:

Find the midpoint of

:

Show

is on

:

Therefore

is a perpendicular bisector of

.

A Theorem from Elementary Geometry States: If P1 and P2

A Theorem from Elementary Geometry States: If P₁ and P₂