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 start with points \( C \) and \( D \), first, we need to draw a line through \( C \) and \( D \) (step E). Then, construct circles centered at \( C \) (step B) and \( D \) (step C) (since we start with \( C \) and \( D \), these circles would intersect at points, say \( A \) and \( B \), so we label those intersection points (step D). Then, drawing a line through \( A \) and \( B \) (step F) is part of the construction. Constructing a circle centered at \( A \) (step A) is not part of starting with \( C \) and \( D \) initially.

So the correct steps are B, C, D, E, F.

• Given \( A \) and \( C \) are centers of circles. Let's assume the circles are constructed such that \( AB \), \( AC \), \( AD \) (if \( D \) is on the circle centered at \( A \)) and \( BC \), \( CD \) (if \( C \) is center) have certain equalities.
• For option A: If \( A \) and \( C \) are centers, and \( B \) is a point on both circles (since it's a compass-straightedge construction, likely \( AB = AC \) (radii of circle centered at \( A \) and \( C \) respectively? Wait, no, if \( A \) is center of one circle and \( C \) is center of another, and \( B \) is on both circles, then \( AB \) is radius of circle \( A \), \( BC \) is radius of circle \( C \). If the circles are constructed with equal radii (common in constructions like perpendicular bisector), then \( AB = BC \) (since \( AB \) and \( BC \) are radii of circles centered at \( A \) and \( C \) with same radius).
• Option B: \( AB \) and \( BD \): If \( D \) is on the circle centered at \( A \), then \( AB = AD \), but \( BD \) would be a chord, not necessarily equal to \( AB \) unless specific construction, so B is false.
• Option C: \( AD = 2AC \): If \( AC \) is the distance between centers, and \( AD \) is diameter (if \( C \) is midpoint), but not necessarily. Wait, if the circles are constructed with \( AC \) as radius, then \( AD \) (if \( D \) is on the circle centered at \( A \) with radius \( AC \)) would be \( AC \), not \( 2AC \). Wait, maybe the diagram is of two circles intersecting or with \( A \) and \( C \) centers, and \( B \), \( D \) points. Let's re - evaluate. If \( A \) is center, \( AB \) and \( AD \) are radii, so \( AB = AD \). If \( C \) is center, \( CB \) and \( CD \) are radii, so \( CB = CD \). If the circles are constructed with \( AB = AC \) (same radius), then:
• Option A: \( AB = BC \): Since \( AB \) is radius of circle \( A \), \( BC \) is radius of circle \( C \), if radii are equal, \( AB = BC \), so A is true.
• Option B: \( AB = BD \): Not necessarily, unless \( D \) is such that \( AB = BD \), but no info, so B false.
• Option C: \( AD = 2AC \): If \( AC \) is the distance between centers, and \( AD \) is a radius? No, if \( A \) is center, \( AD \) is radius, \( AC \) is distance between centers. If \( AC \) is equal to radius (e.g., in construction of perpendicular bisector, where \( AC = AB = BC \)), then \( AD \) (if \( D \) is on the circle centered at \( A \) with radius \( AC \)) would be \( AC \), not \( 2AC \). Wait, maybe the diagram has \( A \) and \( C \) with \( AC \) as radius, and \( D \) is such that \( AD = 2AC \). Wait, maybe I made a mistake. Let's think again. If \( A \) is center, and \( C \) is on the circle centered at \( A \), then \( AC = AB = AD \) (radii). Then \( AD = AC \), not \( 2AC \). So C is false.
• Option D: \( BC = CD \): Since \( C \) is center, \( BC \) and \( CD \) are radii of circle centered at \( C \), so \( BC = CD \), so D is true.
• Option E: \( BD = CD \): \( CD \) is radius, \( BD \) is a chord, not necessarily equal, so E is false.

So the true statements are A, D.

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 \( B \), E. Draw a line through points \( C \) and \( D \), F. Draw a line through points \( A \) and \( B \)

Question 5