Unit 7 - Congruence, Construction, and Proof - ALA
Constructions - Unit 7 - Aligned with MVP
Table of Contents
- Introduction
- Introduction to Geometry Tools
- Unit 7 - 1
- Unit 7 Lesson 1: Under Construction
- Unit 7 Lesson 1 Homework: Under Construction
- Unit 7 - 2
- Unit 7 Lesson 2: More Things Under Construction
- Unit 7 Lesson 2 Homework: More Things Under Construction
Introduction to Geometry Tools
Online Geometry Tools
Welcome to GeoGebra! This will be your online tool for Unit 7. We will be completing lesson notes and the homework in this software. [br][br]Today you will be learning how to use GeoGebra for geometric constructions.
Task 1
Draw a line segment in the space below.
Task 2
Draw different sized circles in the space below.[br][br]Try different circle tools.
Task 3
Draw congruent circles in the space below.[br][br]Try different circle tools.
Task 4
Draw concentric circles in the space below.[br][br]Try different circle tools.
Task 5
Draw a segment congruent to the given segment [math]AB[/math].
Task 6
Draw the perpendicular bisector of the segment [math]RS[/math] shown below.
Task 7
Create an isosceles triangle in the space below.
Unit 7 Lesson 1: Under Construction
Explore
[b]Construct a Rhombus[br][/b][br]Knowing what you know about circles and line segments, how might you locate point [math]D[/math] on the ray in the diagram given, so the distance from [math]B[/math] to [math]D[/math] is the same as the distance from [math]B[/math] to [math]A[/math]?
1. Describe how you will locate point [math]D[/math] and how you know [math]BD\cong BA[/math], then construct point [math]D[/math] on the diagram given.
Now that we have three of the four vertices of the rhombus, we need to locate point [math]E[/math], the fourth vertex.[br][br]2. Describe how you will locate point [math]E[/math] and how you will know [math]DE\cong EA\cong AB[/math], then construct point [math]E[/math] on the diagram.
[b]Construct a Perpendicular Bisector of a Segment and a Square (a rhombus with right angles)[br][br][/b]The only difference between constructing a rhombus and constructing a square is that a square contains right angles. Therefore, we need a way to construct perpendicular lines using only a compass and a straight edge.[br][br]We will begin by inventing a way to construct a perpendicular bisector of a line segment.
Describe your strategy for locating points on the perpendicular bisector of [math]RS[/math].
Now that you have created a line perpendicular to [math]RS[/math], we will use the right angle formed to construct a square. [br][br]Label the midpoint of [math]RS[/math] on the diagram as point [math]C[/math]. Using [math]RC[/math] as one side of the square, and the right angle formed by [math]RC[/math] and the perpendicular line drawn through the point [math]C[/math] as the beginning of the square, finish constructing this square on the diagram below. (Hint: Remember that a square is also a rhombus and you have already constructed a rhombus in the first part of the lesson.)
Takeaways
I used circles and lines as construction tools today.[br][br]Take notes as appropriate below.
Circles are useful construction tools because
Congruent circles are useful construction tools because
The congruent circles in the construction of the perpendicular bisector helped me to notice that
Unit 7 Lesson 2: More Things Under Construction
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 [math]ST[/math] as one side of the equilateral triangle.
Launch
Because [b]regular polygons[/b] have rotational symmetry, they can be [b]inscribed [/b]in a circle. [br][br]The circumscribed circle has its center at the center of rotation and passes through all of the vertices of the regular polygon.[br][br]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.
[list=a][*]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.[br][/*][/list]
Explore
[b]Constructing a Regular Hexagon Inscribed in a Circle[/b]
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 [b]chords[/b] 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?)
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.
Modify your work with the hexagon to construct an equilateral triangle inscribed in the given circle.
It is often useful to be able to construct a line parallel to a given line through a point. For example, suppose we want to construct a line parallel to [math]AB[/math] thought point [math]C[/math] on the diagram below. Since we have observed that parallel lines have the same slope, the line through point [math]C[/math] will be parallel to [math]AB[/math] only if the angle formed by the line and [math]BC[/math] is congruent to angle [math]ABC[/math]. Can you describe and illustrate a strategy that will construct an angle with the vertex at point [math]C[/math] and a side parallel to [math]AB[/math]?
Takeaways
Strategies for constructing geometric figures:
What can we do to construct geometric figures?