Question

4 from unit 1, lesson 2 this diagram was created by starting with points c and d and using only a straightedge and compass to construct the rest. all steps of the construction are visible. select all the steps needed to produce this diagram. a construct a circle centered at a. b construct a circle centered at c. c construct a circle centered at d. d label the intersection points of the circles a and b. e draw a line through points c and d. f draw a line through points a and b. 5 from unit 1, lesson 1 this diagram was constructed with straightedge and compass tools. a is the center of one circle, and c is the center of the other. select all true statements. a ab = bc b ab = bd c ad = 2ac d bc = cd e bd = cd

Explanation:

Question 4

• To construct the diagram starting with points \( C \) and \( D \):
• First, construct circles centered at \( C \) (option B) and \( D \) (option C) with radius equal to the distance between \( C \) and \( D \) (or such that they intersect).
• Then, label the intersection points of these circles as \( A \) and \( B \) (option D).
• Finally, draw a line through \( A \) and \( B \) (option F).
• Option A is incorrect because we start with \( C \) and \( D \), not \( A \). Option E is incorrect as we don't need to draw a line through \( C \) and \( D \) (the circles are centered at them, but the line through \( A \) and \( B \) is the perpendicular bisector).

• Given \( A \) and \( C \) are centers of two circles:
• \( AB \) is a radius of the circle centered at \( A \), \( BC \) is a radius of the circle centered at \( C \). If the circles are constructed with the same radius (since they intersect and \( A \), \( C \) are centers), \( AB = BC \) (option A is false? Wait, no—wait, \( AB \) is radius of \( A \)'s circle, \( BC \) is radius of \( C \)'s circle. If the circles are constructed with \( AC \) as radius? Wait, looking at the diagram, \( A \) and \( C \) are centers, so \( AB = AD \) (radii of \( A \)'s circle), \( BC = CD \) (radii of \( C \)'s circle). Also, \( AD = AC + CD \), but if \( AC = CD \) (since \( C \) is center, \( AC \) could be equal to \( CD \) if \( D \) is on \( A \)'s circle? Wait, let's re - evaluate:
• Option B: \( AB \) and \( BD \): \( AB \) is radius of \( A \)'s circle, \( BD \): Is \( BD \) equal to \( AB \)? If \( D \) is on \( A \)'s circle, then \( AD = AB \) (no, \( AD \) would be a chord). Wait, the correct approach:
• Since \( A \) is center, \( AB = AD \) (radii). \( C \) is center, so \( BC = CD \) (radii, so option D is true). Also, \( AD = AC + CD \), and if \( AC = CD \) (since \( C \) is between \( A \) and \( D \) as per diagram), then \( AD = 2AC \) (option C is true). \( AB = AD \) (radius of \( A \)'s circle), and \( BD \): Wait, \( AB = BC \) (if \( AC = AB \))? No, let's look at each option:
• Option A: \( AB = BC \): \( AB \) is radius of \( A \)'s circle, \( BC \) is radius of \( C \)'s circle. If the circles are constructed with \( AB = BC \) (since they intersect at \( B \)), but is this necessarily true? Wait, no—wait, the diagram shows that \( A \) and \( C \) are centers, and \( B \) is the intersection. So \( AB \) (radius of \( A \)) and \( BC \) (radius of \( C \)): if the circles are drawn with radius equal to \( AC \), then \( AB = AC \) and \( BC = AC \), so \( AB = BC \) (option A is true? Wait, maybe I made a mistake earlier. Let's start over:
• Let's assume the circles are constructed with radius equal to the distance between \( A \) and \( C \). Then:
• \( AB \): radius of \( A \)'s circle, so \( AB = AC \).
• \( BC \): radius of \( C \)'s circle, so \( BC = AC \). Thus, \( AB = BC \) (option A is true).
• \( AB = BD \): \( BD \): Is \( BD = AB \)? \( BD \) is a chord of \( A \)'s circle? No, \( D \) is on \( A \)'s circle (since \( AD \) is a radius? Wait, \( A \) is center, so \( AD \) is a radius, so \( AD = AB \). Then \( BD \): In triangle \( ABD \), if \( AD = AB \), it's isoceles, but is \( BD = AB \)? No, unless it's equilateral. So option B is false.
• \( AD = 2AC \): \( AD = AC + CD \), and \( CD = BC = AC \) (since \( BC = AC \) and \( CD = BC \) as \( C \) is center), so \( AD = AC + AC = 2AC \) (option C is true).
• \( BC = CD \): \( BC \) and \( CD \) are both radii of the circle centered at \( C \), so \( BC = CD \) (option D is true).
• \( BD = CD \): \( CD = BC \), and \( BD \): Is \( BD = BC \)? In triangle \( BCD \), if \( BC = CD \), it's isoceles, but \( BD \) is not necessarily equal to \( CD \) unless it's equilateral. So option E is false.

Answer:

B. Construct a circle centered at \( C \), C. Construct a circle centered at \( D \), D. Label the intersection points of the circles \( A \) and \( B \), F. Draw a line through points \( A \) and \( B \)

Question 5