Mathematics
Grade 10
15 min
Construct the midpoint or perpendicular bisector of a segment (Tutorial Only)
Construct the midpoint or perpendicular bisector of a segment (Tutorial Only)
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1
Introduction & Learning Objectives
Learning Objectives
Define a perpendicular bisector and a midpoint.
Identify and properly use a compass and straightedge for geometric constructions.
Follow the step-by-step process to construct the perpendicular bisector of any given line segment.
Use the construction to locate the exact midpoint of a line segment.
Explain the geometric principles that prove the construction is valid.
Verify that the constructed line is both perpendicular to and bisects the original segment.
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This tutorial will guide you through the classical geometric construction of a perpendicular bisector using only a compass and a straightedge. You'll l...
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Key Concepts & Vocabulary
TermDefinitionExample
Line SegmentA part of a line that is bounded by two distinct end points and contains every point on the line between its endpoints.The edge of a book is a line segment. In geometry, we denote a segment with endpoints A and B as \(\overline{AB}\).
MidpointThe point on a line segment that divides it into two segments of equal length.If \(\overline{AC}\) is 10 cm long, its midpoint B is at the 5 cm mark, making \(\overline{AB}\) and \(\overline{BC}\) both 5 cm long.
Perpendicular LinesTwo lines that intersect to form a right angle (90°).The corner of a square is formed by two perpendicular line segments.
BisectorA line, ray, or segment that passes through the midpoint of another segment, dividing it into two equal parts.A fold line down the exact center of a rectangular...
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Core Formulas
Perpendicular Bisector Theorem
If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
This is the core principle behind our construction. By creating intersection points that are the same distance from both endpoints of the segment, we guarantee that the line connecting them is the perpendicular bisector.
Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.
This theorem proves our construction works. The points where our arcs intersect are, by definition, equidistant from the segment's endpoints. Therefore, they must lie on the perpendicular bisector.
Definition of a Midpoint
If M is the midp...
4 more steps in this tutorial
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Challenging
The construction process creates two points, R and S, such that PR = QR and PS = QS. Which geometric principle is the primary justification for concluding that the line passing through R and S *must* be the perpendicular bisector of PQ?
A.The Converse of the Perpendicular Bisector Theorem.
B.The Pythagorean Theorem.
C.The definition of a midpoint.
D.The SSS Triangle Congruence Postulate.
Challenging
A student argues, 'The construction is valid because it creates two congruent triangles on either side of the segment, and congruent triangles have equal heights, which proves the line is a bisector.' Why is this reasoning incomplete for proving the construction of a *perpendicular* bisector?
A.The reasoning is completely correct and sufficient.
B.It incorrectly assumes the triangles are congruent; they are only similar.
C.It fails to address the 'perpendicular' requirement, which involves proving a 90-degree angle is formed.
D.It only proves the line is a bisector, but not that it passes through the midpoint.
Challenging
During the construction on segment XY, you place the compass on X and draw an arc, then on Y and draw a second arc. Why is it a logical necessity to use the *exact same radius* for both arcs?
A.It is not necessary, any two radii larger than half the segment length will work.
B.To ensure the intersection points are equidistant from X and Y, which is the condition required by the Converse of the Perpendicular Bisector Theorem.
C.To make the two large triangles (e.g., ΔXRY and ΔXSY) right triangles.
D.To ensure the arcs are perfectly symmetrical for aesthetic reasons.
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