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G.7(A)·Geometry·Coordinate GeometrySupporting

Perpendicular Bisectors

Use coordinates to analyze perpendicular bisectors.

Students locate and write perpendicular bisectors on the coordinate plane by combining the midpoint with the perpendicular slope, and verify that points on the bisector are equidistant from the endpoints.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutorAll Geometry released practice50 minutes

Learn the concept

Two requirements

A perpendicular bisector passes through a segment's midpoint and meets the segment at a right angle. Both conditions are required.

Build the equation

Find the midpoint by averaging coordinates. Find the segment's slope, then use the negative reciprocal: a slope of 23\frac{2}{3} becomes −32-\frac{3}{2}. Write the line with point-slope form using the midpoint.

Equidistance

Every point on a perpendicular bisector is the same distance from both endpoints. A horizontal segment has a vertical bisector, x = midpoint x-coordinate; a vertical segment has a horizontal bisector.

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Geometry EOC · 2013 · Item 10 · 1 point

On a coordinate grid from negative 5 to 5, horizontal segment PQ runs from P at negative 4, 2 to Q at negative 1, 2.
Segment PQ on a coordinate gridOriginal visual from the released STAAR test.

Worked example

P(−2, 1) and Q(4, 5) are endpoints. Find the point on the perpendicular bisector of PQ with y = 0.

  1. The midpoint is (1, 3).
  2. The slope of PQ is 46=23\frac{4}{6}=\frac{2}{3}, so the bisector slope is −32-\frac{3}{2}.
  3. The bisector is y−3=−32(x−1)y-3=-\frac{3}{2}(x-1). Substitute y=0y=0: −3=−32(x−1)-3=-\frac{3}{2}(x-1), so x=3x=3.
  4. Check: the distances from (3,0)(3,0) to P and Q are both 26\sqrt{26}.

(3, 0)

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Geometry - Determining Perpendicular Lines

Five Minute Math · Aaron Daffern, Dallas ISD

Use negative-reciprocal slopes to write lines perpendicular to a given line.

Watch on YouTube
Texas public school video

Geometry - Using the Midpoint Formula

Five Minute Math · Aaron Daffern, Dallas ISD

Find the midpoint that a perpendicular bisector must pass through.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

  1. 1. Write the perpendicular bisector of the segment from (2, 3) to (10, 3).

    Show answer and reasoning

    x = 6

    The segment is horizontal with midpoint (6, 3), so its perpendicular bisector is the vertical line through x = 6.

  2. 2. Is (0, 0) on the perpendicular bisector of the segment from A(−1, 3) to B(3, −1)?

    Show answer and reasoning

    Yes

    The distance from (0,0)(0,0) to each endpoint is 10\sqrt{10}, so the point is equidistant from A and B.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 10 · 1 point

On a coordinate grid from negative 5 to 5, horizontal segment PQ runs from P at negative 4, 2 to Q at negative 1, 2.
Segment PQ on a coordinate gridOriginal visual from the released STAAR test.

PQ‾\overline{PQ} is shown on the coordinate grid. The coordinates of PP and QQ are integers. Point (x,y)(x, y) lies on the perpendicular bisector of PQ‾\overline{PQ}. What is the value of xx? Record your answer.

Show STAAR answer and explanation

-2.5

P=(−4,2)P=(-4,2) and Q=(−1,2)Q=(-1,2). The midpoint is ((−4−1)/2,2)=(−2.5,2)((-4-1)/2,2)=(-2.5,2). Because PQ is horizontal, its perpendicular bisector is the vertical line x=−2.5x=-2.5.

-1.5: The midpoint's x-coordinate is (−4+(−1))/2=−2.5(-4+(-1))/2=-2.5, not −1.5.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

  • Find the midpoint and slope of a segment on the coordinate plane.
  • Write the equation of a segment's perpendicular bisector.
  • Use the equidistance property to check points on a perpendicular bisector.

Instructional sequence

  1. 1

    Engage

    5 min

    Fold a segment drawn on patty paper so its endpoints match. Ask what the crease shows.

  2. 2

    Direct instruction

    10 min

    Name the two requirements: through the midpoint and perpendicular to the segment.

  3. 3

    Guided practice

    15 min

    Find midpoints and negative-reciprocal slopes, then write point-slope equations.

  4. 4

    Equidistance check

    12 min

    Test points on and off the bisector with the distance formula.

  5. 5

    Closure

    8 min

    Complete the exit ticket and verify one point on the bisector.

Misconceptions to address

““A perpendicular bisector has the same slope as the segment.””

Teacher move: Equal slopes make parallel lines. A perpendicular line uses the negative reciprocal slope.

““The bisector passes through an endpoint.””

Teacher move: Bisect means divide into two equal parts, so the line must pass through the midpoint.