**09.03 Module Nine Quiz**

Step 2: Continuation

Question 3: The team thinks the black bears are at the position of point F. The rangers use two-way 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.

According to the Exterior Angle to a Circle Theorem, (Arc #2 - Arc #1) / 2 = Angle. So (250 - 110) / 2 = 70. " The radios from the ranger station can only receive signals within a 65° angle before the signal is lost." Since the ranger station is 70°, they can't communicate with both teams. They will risk losing the signal. Segments DE and DF are both tangent to the circle. They are tangent, because they touch the outside of the circle and don't go through.

Step 1: Constructions

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.

The central angle is 120°. This means that the arc BEC would be it's intercepted arc. An intercepted angle and a central angle both have the same value. So arc BEC is 120° too.

Step 2: Continuation

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.

If <ACB is 30° and it is also an inscribed angle, then you can use the Inscribed Angle Theorem to find the arc that it intercepts. The theorem would be <ACB = 1/2 arc. Which is 30° = 1/2 arc. So the inscribed angle is equal to half of the intercepted arc. Therefore, the intercepted arc would be 60°.

In order to find the inscribed circle, first I found the points of intersection. The first point is on segment AB and the second on segment CB. And from those two points, I found the point on segment AC. I connected this point to point B. Then I drew 2 additional points on segment AC and BC in order to find a point that will go through point C and the point on segment AB. And in order to find the circumscribed circle, I used the center, which I found when I drew the inscribed circle.

120°

120°

30°

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.

The scale factor, K= r2 / r1 is used for concentric circles. If this scale factor is used for these circles, it would be b / a. Even though the radius of these two circles are different they are still congruent. And these circles are similar since they both share the same triangle.

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?

Hint:

Will the angle increase or decrease in size as point D moves along the circle upwards towards the x-axis? What about as D moves downward towards the y-axis? What will happen to the angle? Use the diagram below for help. You must show all work and provide an explanation to receive credit.

If point D is moved along the edge of the circle towards the x-axis, then angle ACD will decrease in size. And if point D is moved towards the y-axis, then angle ACD will also decrease in size.

Step 3: Answering the Questions

110°

250°

Circumscribed Circle

Inscribed Circle