09.03 Module Nine Quiz
Step 1: Constructions
Explain using words how you constructed the circumscribed circle and inscribed circle and what steps were
involved in the process. You must show all work to receive credit.
To find the circumscribed circle, I first had the find the perpendicular bisectors of two of the sides. So I
opened up my compass a struck arcs above a below side AB, first from vertex A then from vertex B. Then I
struck arcs above and below line BC, first from vertex B and then from vertex C. After drawing a line through
where the arcs meet above and below each of the sides, I found the circumcenter. I then opened up my
compass so that it was the width from the circumcenter to one of the vertices and drew a circle around the
triangle that intersected all three of the vertices.
To find the inscribed circle, I first found two of the angle bisectors. I opened up my compass and placed it
first on A and then on C, and drew arcs that intersected sides AB, AC, and BC. Then I found the angle
bisector of A by placing my compass on the points on sides AB and AC and striking two arcs that intersect
each other. Then I used by straight edge to draw a line from point A, through where the two arcs intersect.
Then I found the angle bisector of B by placing my compass on the points on sides BC and AC and striking
two arcs that intersect each other. Then I used by straight edge to draw a line from point C, through where
the two arcs intersect. The two straight lines I just drew intersect each other at the incenter of the triangle.
From here, I open up the compass to the width from where the lines intersect to a point on the triangle and
swing a circle that stays within the triangle to create the inscribed circle.
Step 2: Relationships
Question 1: Campsite #1, Lookout Tower, and campsite #2, form a central angle within the
circle. If the angle formed is 120°, describe the relationship between the angle and the arc it
intercepts. You must show all work to receive credit. Because the angle created is a central angle,
the angle and the arc it intercepts have the same measure. So the intercepted arc has a measure of 120
degrees.
Question 2: An inscribed angle is formed by Lookout Tower, campsite #1, and campsite #2. If
the angle formed is 30 degrees, describe the relationship between the angle and the arc it
intercepts along the circle. You must show all work to receive credit. Since an inscribed angle is
half the measurement of the arc it intersects, we know that the intersected arc must have a measure of 60
degrees.
Question 3: The team thinks the black bears are at the position of point F. The rangers use twoway radios to communicate with one another. The radios from the ranger station can only
receive signals within a 65° angle before the signal is lost. A ranger team is located at point E
while another team heads to point F to attempt to lure the bears to a more remote location. If
the arc formed by ECF is 110°, can the ranger station communicate with both teams, or will they
risk losing the signal? What description can be given to the line that connects the ranger station
with either point E or point F and how do you know? You must show all work and steps to
receive credit. The two lines that connect the ranger station to point E and point F can be classified as
tangents because they both intersect the circle at just one point. The ‘Exterior Angle to a Circle Theorem’ states
that if two secants, two tangents, or a secant and a tangent intersect outside a circle, the measure of the created
angle between them is one-half the absolute value of the difference of the measures of their intercepted arcs.
This means that we can subtract the measure of arc ECF (110) from the measure of arc EBF (250) to get 140
degrees, then divided it to find that the measure of angle EDF (the ranger station) is 70 degrees. Because this is
more than the 65 degrees measure that the radios work within, the ranger stations risks losing the signal.
Step 3: Answering the Questions
Question 1: For the constructions completed in step 1, let the radius of the inscribed circle be
A and the radius of the circumscribed circle be b. Show that the two circles constructed (the
circumscribed circle and the inscribed circle) are similar using the radius for each. You must
show all work to receive credit. Since the radius is the same from any point of the circle, the ratio
created by the radius of two circles will be the same from any point of the circle, which means they have the
same shape and are similar. So this means that a over b will be equal no matter where the measure of the
ratio is taken from the circle, so the two circles are similar.
Question 2: In the diagram below, how does angle ACD change as you move the point of
tangency (point D) along the edge of the circle? If the point was moved along the arc of the circle, not only
would the line turn into a secant instead of a tangent, the angle would decrease according to where along the circle the
point was moved. If it is moved closer to the x-axis, the angle would decrease and if it is moved closer to the y-axis the
angle would increase.