Question

Determine whether point T lies on the perpendicular bisector of N•

T(6, 2) N(1, 1) P(5, 7)
• Use your Geometry notebook to answer the problem.
• Show your work, explain your reasoning, and then upload a clean image of your work here for your instructor to grade.
Up to 7 points for showing the correct answer and the steps you took to arrive at that answer (up to 5 points for either correctly verifying perpendicularity or the midpoint or 7 points for verifying both).
• Up to 3 points for stating the correct postulate or explaining the correct reasoning

Answer 1

T lies on the perpendicular bisector of NP

Explanation:

You want to know if point T(6, 2) lies on the perpendicular bisector of segment NP with end points N(1, 1) and P(5, 7).

Graph

The attachment shows a graph of the points in the problem. It also shows the perpendicular bisector of NP and the fact that T lies on the perpendicular bisector.

Theorem

While the geometry application demonstrates the answer, we can also figure it by looking at the distances NT and PT. Each of those distances is the hypotenuse of a right triangle with legs 1 unit and 5 units. Those right triangles are LL congruent, so their hypotenuses are the same length (CPCTC).

The Perpendicular Bisector Theorem states that a point that is equidistant from the end points of a segment lies on its perpendicular bisector. We have shown that T is equidistant from N and P, so T lies on the perpendicular bisector of NP.

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Additional comment

The midpoint M of NP is ((1, 1) +(5, 7))/2 = (3, 4).

The vector NP is P -N = (5, 7) -(1, 1) = (4, 6).

The vector MT is T -M = (6, 2) -(3, 4) = (3, -2).

The dot product of these vectors is 0 when they are perpendicular:

NP·MT = (4, 6)·(3, -2) = 4·3 +6(-2) = 12 -12 = 0

The line MT goes through the midpoint of NP and is perpendicular to NP, so T lies on the perpendicular bisector of NP.

There are other ways you can show this, too. For example, you could write the equation for the line that is perpendicular to NP through its midpoint M, then show that T satisfies the equation for that line.

2x +3y = 18

2(6) +3(2) = 18, so T is on the bisector

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