Question

lesson 2
constructing patterns
lets use compass and straightedge constructions
to make patterns.
warm - up math talk: why is that true?
here are 2 circles with centers a and b.
based on the diagram, decide whether each statement is true. be prepared to share
your reasoning.
1 the length of segment ea is equal to the length of segment eb.
2 triangle abf is equilateral.
3 ( ab=\frac{1}{2}cd )
4 ( cb = da )

Explanation:

1.

Step1: Analyze the length of \(EA\) and \(EB\)

\(EA\) is the radius of the circle with center \(A\), and \(EB\) is the radius of the circle with center \(B\). Since the two circles are not the same size (as can be seen from the diagram where the circles have different extents relative to the line \(CD\)), \(EA
eq EB\).

2.

Step1: Check the sides of \(\triangle ABF\)

\(AB\) is the distance between the centers of the two circles. \(AF\) is the radius of the circle with center \(A\), and \(BF\) is the radius of the circle with center \(B\). If we assume the radius of the circle with center \(A\) is \(r_A\) and with center \(B\) is \(r_B\), and from the construction (using compass - straightedge, if the circles are constructed such that \(r_A = AB=r_B\)) (because when we construct with a compass, if we set the compass width to \(AB\) to draw both circles), then \(AB = AF=BF\). So, \(\triangle ABF\) is equilateral.

3.

Step1: Express \(CD\) in terms of \(AB\)

Let \(r\) be the radius of the circle with center \(A\) (and also the radius used to construct the circle with center \(B\) relative to the line \(CD\)). \(CD=CA + AB+BD\). Since \(CA = r\), \(AB = r\), \(BD = r\) (assuming the compass - constructed circles where the radius is set to \(AB\)), then \(CD=3r\) and \(AB = r\), so \(AB
eq\frac{1}{2}CD\).

4.

Step1: Express \(CB\) and \(DA\) in terms of \(AB\)

\(CB=CA + AB\) and \(DA=AB + BD\). Since \(CA = BD\) (radii of the same - sized circles constructed with the compass width set to \(AB\)), then \(CB=DA\).

Answer:

1. False.
2. True.
3. False.
4. True.