Curriculum  /  Math  /  10th Grade  /  Unit 1: Constructions, Proof, and Rigid Motion  /  Lesson 5

Constructions, Proof, and Rigid Motion

Lesson 5

Math

Unit 1

10th Grade

Lesson 5 of 19

Objective

Construct the altitudes and perpendicular bisectors of sides of triangles.

Common Core Standards

Core Standards

  • G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
  • G.CO.D.12 — Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Foundational Standards

  • 6.G.A.1

Criteria for Success

  1. Construct the perpendicular bisectors of each side of a triangle. 
  2. Define an altitude as a line segment drawn from a vertex perpendicular to the opposite side (or an extension of the opposite side) and use the definition in the solution of problems.
  3. Describe the features of triangles where the altitude of a triangle and the perpendicular bisector of the triangle are the same line segment.

Tips for Teachers

  • Students will find the orthocenter (the point of concurrency where the altitudes of a triangle intersect), and the circumcenter (the point of concurrency where the perpendicular bisectors of a triangle intersect) but they will not need to memorize these terms. All the points of concurrency will be explored again in Unit 2. 
  • Geogebra “Points of Concurrency” may be helpful to investigate how the points of concurrency change based on the angle measures of a triangle.
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Anchor Problems

25-30 minutes

Problem 1

Construct the perpendicular bisectors of the sides of the triangle.

 

Guiding Questions

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Problem 2

In a triangle, an altitude is a line segment drawn from a vertex perpendicular to the opposite side (or an extension of the opposite side). 

Use this information to construct the altitudes of the triangle.

Guiding Questions

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Problem 3

Jamie makes the statement “the point of concurrency formed by the altitudes of an equilateral triangle is the same as the point of concurrency formed by the perpendicular bisectors of an equilateral triangle.”  

Is Jamie’s statement correct? Explain your reasoning.

Guiding Questions

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Target Task

5-10 minutes

Use your knowledge of altitudes and perpendicular bisectors to create an equilateral triangle using $${\overline{AB}}$$ as a side length. Describe the relationship between the perpendicular bisectors and the altitudes of the triangle you have constructed.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include additional application problems that will help students see the usefulness of finding the points of concurrency formed by the perpendicular bisectors and the altitudes.

Next

Use principles of proof to justify each step in solving an equation.

Lesson 6
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Lesson Map

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Topic A: Constructions of Basic Geometric Figures

Topic B: Justification and Proof of Angle Measure

Topic C: Translations of Points, Line Segments, and Angles, and Parallel Line Relationships

Topic D: Reflections and Rotations of Points, Line Segments, and Angles

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