Lesson 4 Circles Inside Out Solidify Understanding

Jump Start

You worked on the following problem as the Exit Ticket for the previous lesson.

Given: $\overset{―}{A B}$ is a diameter of the circle shown in the diagram.

1.

What observations can you make about the three inscribed angles, $∠ A C B$ , $∠ A D B$ , and $∠ A E B$ ?

2.

State a theorem based on these observations.

Learning Focus

Examine the relationships between circumscribed angles and circles.

We have seen that circles can be inscribed in triangles so that the circle is tangent to all three sides. What relationship exists between the radii of the inscribed circle and the sides of the triangle that are tangent to the circle?

How do I construct lines that are tangent to a circle from a point that lies outside the circle?

How do I measure a circumscribed angle formed by two intersecting lines tangent to a circle?

Open Up the Math: Launch, Explore, Discuss

In Centers of a Triangle, your work on Kolton’s notes and diagram should have convinced you that it is possible to locate a point that is equidistant from all three sides of a triangle, and therefore, a circle can be inscribed inside every triangle.

1.

Based on Kolton’s work, use a compass and straightedge to construct the circles that can be inscribed in each of the following triangles.

2.

Once you have located the center of the inscribed circle, how do you determine where the points of tangency between the circle and the sides of the triangle are located?

To construct an inscribed circle in problem 1, you not only had to locate the center of the inscribed circle using the angle bisectors, but you also had to locate the points on the sides of the triangles that were equidistant from the center. By either experimentation or reasoning you should have found that these points would lie on the lines perpendicular to the sides of the triangle through the center of the circle. We will use this observation for the remainder of the lesson. (If you have not already convinced yourself that the following statement is true, continue to think about how you would justify this statement as you work on the remainder of the Exploration task.)

Theorem: A tangent line to a circle is perpendicular to the radius at the point of tangency.

3.

Angles formed by lines that are tangent to a circle are called circumscribed angles. Use a protractor to measure the circumscribed angles relative to the arcs they intercept for the three triangles in problem 1. Make a conjecture about the measures of the circumscribed angles. Then prove your conjecture using what you know about inscribed and central angles.

My conjecture about the measures of circumscribed angles:

Proof of my conjecture:

What else can we learn about the quadrilateral whose vertices consist of the center of the circle, the two points of tangency, and the vertex of the circumscribed angle, as shown in the diagram?

4.

Prove: $\overset{―}{A B} ≅ \overset{―}{C B}$

Quadrilateral $A B C D$ is an example of a type of quadrilateral called a kite.

5.

Prove that diagonal $\overset{―}{B D}$ bisects opposite angles $∠ A B C and ∠ A D C$ .

6.

Prove that the diagonals of a kite are perpendicular to each other.

Ready for More?

Based on your work in this task and the previous task, describe a procedure for constructing a tangent line to a circle through a given point outside the circle. Can you prove that your procedure works?

Once you have a strategy, work with a partner to revise and refine your proof of why your procedure to construct a tangent line to a circle works.

Takeaways

Theorem: Angles inscribed in a half-circle (or semi-circle) , because

Angle $B$ circumscribes circle $D$ if:

The measure of a circumscribed angle is:

The tangent segments, which are drawn from to are .

Quadrilateral $A B C D$ is a kite. A quadrilateral is a kite if it satisfies the following conditions:

The diagonals of a kite are

Vocabulary

angles associated with circles: central angle, inscribed angle, circumscribed angle
kite
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about a class of quadrilaterals that are called kites. Kites occurred in our work when a circle was inscribed within a triangle. The segments drawn from the vertex of the circumscribed angle to the points of tangency on the circle, along with the two radii of the circle drawn to the points of tangency, form the sides of the kite. By examining features of the kite, we were able to develop a formula for the measure of the circumscribed angle, relative to the intercepted arc.

Retrieval

1.

Draw the altitude from vertex $A$ in triangle $C A B$ . Then use trigonometry to find the height of the altitude. Leave your answer in terms of $\sin θ$ , $\cos θ,$ or $\tan θ$ .

(Hint: Answer will look something like this: $H = n \sin θ$ )

2.

Write the trigonometric equation needed to solve for angle $x$ . Then solve for $x$ .