Lesson 2 More Things Under Construction Solidify Understanding

Jump Start

Like a rhombus, an equilateral triangle has congruent sides. Show and describe how you might locate the third vertex point on an equilateral triangle, given $\overset{―}{S T}$ as one side of the equilateral triangle.

Learning Focus

Construct parallel lines and inscribed regular polygons.

How do I use geometric objects, such as circles and lines, to construct geometric figures like parallel lines and regular polygons, rather than using measurement tools, such as rulers and protractors, to draw such figures?

Open Up the Math: Launch, Explore, Discuss

Because regular polygons have rotational symmetry, they can be inscribed in a circle. The circumscribed circle has its center at the center of rotation and passes through all of the vertices of the regular polygon.

We might begin constructing a hexagon by noticing that a hexagon can be decomposed into six congruent equilateral triangles, formed by three of its lines of symmetry.

1.

Sketch a diagram of such a decomposition.
Based on your sketch, where is the center of the circle that would circumscribe the hexagon?
Use a compass to draw the circle that would circumscribe the hexagon.

Constructing a Regular Hexagon Inscribed in a Circle

2.

The six vertices of the regular hexagon lie on the circle in which the regular hexagon is inscribed. The six sides of the hexagon are chords of the circle. How are the lengths of these chords related to the lengths of the radii from the center of the circle to the vertices of the hexagon? That is, how do you know that the six triangles formed by drawing the three lines of symmetry are equilateral triangles? (Hint: Considering angles of rotation, can you convince yourself that these six triangles are equiangular and therefore equilateral?)

3.

Based on this analysis of the regular hexagon and its circumscribed circle, illustrate and describe a process for constructing a hexagon inscribed in the given circle.

4.

Modify your work with the hexagon to construct an equilateral triangle inscribed in the given circle.

Constructing a Parallel Line Through a Given Point

5.

It is often useful to be able to construct a line parallel to a given line through a given point. For example, suppose we want to construct a line parallel to $\overset{―}{A B}$ through point $C$ on the diagram below. Since we have observed that parallel lines have the same slope, the line through point $C$ will be parallel to $\overset{―}{A B}$ only if the angle formed by the line and $\vec{B C}$ is congruent to $∠ A B C$ . Can you describe and illustrate a strategy that will construct an angle with the vertex at point $C$ and a side parallel to $\overset{―}{A B}$ ?

Ready for More?

Describe how you might construct a square inscribed in a circle. Test out your construction on the circle below.

Takeaways

Strategies for constructing geometric figures:

Vocabulary

circumscribe
inscribed in a circle
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to construct regular hexagons and equilateral triangles by inscribing them in a circle. We also learned how to construct a line parallel to a given line through a given point. These constructions were based on properties of figures that we have observed in previous lessons.

Retrieval

1.

A pair of lines are given; one is the pre-image and the other is the image.

a.

Define the transformation used to create the image.

b.

Write the equation for the graph of each line.

2.

Write an explicit function rule and a recursive function rule for the values in the table.

Step Number: $n$	$1$	$2$	$3$	$4$	$5$
$f (n)$	$35$	$245$	$1,715$	$12,005$	$84,035$