Question

a. make a conjecture about segment ab and segment bc in the below construction that was done with a compass and straightedge

claim

image of two intersecting circles with points a, b, c

b. support that claim with strong evidence (notation, evidence, reasoning).

c. give an example of evidence that ms. sens and mr. stone would consider weak evidence

d. what were the main mathematical concepts or ideas that you learned in this unit?

Explanation:

Part A

From the construction with compass and straightedge, we observe that the circles are constructed such that \( A \) and \( B \) are centers of two circles that intersect at \( C \). So, the radii of the circles ( \( AB \) is a radius of one circle, and \( BC \) is a radius of the other, but actually, since the circles are constructed with \( A \) and \( B \) as centers and equal radii (as they intersect and form this figure), we can conjecture that \( AB = BC \).

• Notation: Let the circle with center \( A \) have radius \( r_A \) and the circle with center \( B \) have radius \( r_B \).
• Evidence: In the construction, the two circles are drawn with a compass such that they intersect at point \( C \). The length \( AB \) is the radius of one circle (say, the circle centered at \( A \) passes through \( B \), and the circle centered at \( B \) passes through \( A \)), and the length \( BC \) is the radius of the circle centered at \( B \) (since \( C \) lies on the circle centered at \( B \)). Also, the circle centered at \( A \) has radius \( AB \) (because \( B \) is on the circle centered at \( A \)) and the circle centered at \( B \) has radius \( AB \) (because \( A \) is on the circle centered at \( B \)). Since \( C \) is on the circle centered at \( B \), the length \( BC \) is equal to the radius of the circle centered at \( B \), which is \( AB \).
• Reasoning: By the definition of a circle, all radii of a given circle are equal. The circle centered at \( A \) has radius \( AB \) (because \( B \) is on the circle), and the circle centered at \( B \) has radius \( AB \) (because \( A \) is on the circle). Since \( C \) lies on the circle centered at \( B \), \( BC \) is a radius of the circle centered at \( B \), so \( BC=AB \).

Weak evidence would be something that is not based on mathematical definitions or properties. For example, saying "Segment \( AB \) and segment \( BC \) look equal when I measure them with a ruler" is weak evidence. This is because a ruler measurement is approximate and depends on the precision of the ruler, and it does not use the mathematical properties of circles (like radii) or the construction with compass and straightedge.

Answer:

Segment \( AB \) and segment \( BC \) are congruent (i.e., \( AB = BC \)).

Part B