Topics
7. Geometry
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United States of America
Grade 7

Investigation: Construct geometric figures

Lesson
Practice
Overview
Activity 1
Activity 2
Activity 3
Activity 4
Reflection

Investigate various geometric constructions and the triangle inequality.

Objectives
  • To construct a line segment, angle, and triangle.
  • To consider how many triangles can be created for a given set of angles.
  • To learn the triangle inequality.
Materials
  • Paper
  • Compass
  • Straight edge
  • Pen or pencil

Construct a line segment

Tools of mathematics
Every profession has particular tools that they use all the time, a chef has their knives, the sewer a machine and the mathematician, well they have their construction tools: a compass, pencil, and straight edge.
Euclid
A great Greek mathematician named Euclid, who is credited to have written the first mathematics textbook over 2000 years ago, went to great lengths to detail many of the mathematical constructions we will look at today. Geometrical constructions were so important to mathematics at the time because most problems were solved graphically, not arithmetically.
Construct a segment
To construct congruent line segments with a compass and straight edge, follow the instructions below.
Steps for construction:
  • Start with the line segment \overline{AB} on the page that you want to copy.
  • Mark a point on the page where you want the start of the copied line segment. Call this point C.
  • Set the compass width to the length of \overline{AB}.
  • Without changing the compass width, move the compass to C.
  • Draw an arc. The endpoint of the new line segment can be anywhere on the arc.
  • Choose a point, D, on the arc and draw \overline{CD} with the straight edge.

Play and pause the video below at each step to help you.

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Investigate
Consider the following questions once you have completed the above procedure.
1.
Can you come up with another method to create a line segment of the same length?

Construct an angle

Congruent angles
As we have seen, congruent means same, so congruent angles are 2 (or more) angles that are exactly the same size. They can be facing in any direction, but if they are the same size then the angles are congruent. You can think of congruent angles as copies of each other.
To construct congruent angles you need a compass and a straight edge.
Steps for construction:
  • Start with the angle \angle BAC on the page that you want to copy.
  • Draw a ray, \overrightarrow{PQ}, on the page.
  • Position the compass on point A, adjust to any width.
  • Draw an arc that crosses both legs of \angle BAC, name the points where the arc intersects the rays F and G.
  • Position the compass on point on point P, and draw another arc. Where it crosses \overrightarrow{PQ} call it M.
  • Measure the distance FG with the compass.
  • Place the compass at point M, and cross the arc. Call this intersection N.
  • Draw in a ray from P, through N. Call it \overrightarrow{PR}.
  • Now you have created \angle RPQ.
  • \angle RPQ = \angle BAC

Play and pause the video below at each step to help you.

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Investigate
Consider the following questions once you have completed the above procedure.
1.
Do the lengths of the encompassing sides of your angle change the angle measurement?

Construct a triangle

We're now going to put everything that we've learned together in order to construct a triangle. Follow the instructions below.
To construct triangles you need a compass and a straight edge.
Steps for construction:
  • Start by constructing a 45 \degree angle, call this angle A.
  • Extend both lines that encompass angle A and form a 90 \degree angle on one of the extending lines, call this angle B.
  • Extend all lines to make a triangle and label the final angle C.
Investigate
Consider the following questions once you have completed the above procedure.
1.
What is the measurement of angle C?
2.
What are the side lengths? Compare them with another classmate's side lengths. What do you notice or wonder?

Triangle side lengths

We're now going to build triangles and consider the side lengths.
Create 3 triangles using the applet below and record the side lengths in the downloable asset chart.

Use the following applet. Drag the sliders to change the value of the 3 side lengths, and move the side lengths to create a triangle.

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Investigate
Consider the following questions once you have completed the above procedure.
1.
Discuss any patterns that you notice between the 3 side lengths of your triangles.
2.
Can you draw a triangle with sides that are 12 \text{ cm}, 6 \text{ cm}, and 4 \text{ cm}? Why or why not?

Answer the questions below after you have completed the activity.

Discussion
1.
How many triangles can be made with the angle measurements 45 \degree, 45 \degree, and 90 \degree?
2.
Using your and another classmate's triangle side lengths from Activity 3, can you determine how many triangles can be made with specified angle measurements?
3.
Using your and another classmate's triangle side lengths from Activity 4, can you determine a rule for how to form a triangle using side lengths?

Outcomes

7.G.A.2

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

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