Lesson 4

Quadrilaterals in Circles

  • Let’s investigate quadrilaterals that fit in a circle.

Problem 1

A quadrilateral \(ABCD\) has the given angle measures. Select all measurements which could come from a cyclic quadrilateral.

A:

angle \(A\) is 90\(^\circ\), angle \(B\) is 90\(^\circ\), angle \(C\) is 90\(^\circ\), and angle \(D\) is 90\(^\circ\)

B:

angle \(A\) is 80\(^\circ\), angle \(B\) is 80\(^\circ\), angle \(C\) is 100\(^\circ\), and angle \(D\) is 100\(^\circ\)

C:

angle \(A\) is 70\(^\circ\), angle \(B\) is 110\(^\circ\), angle \(C\) is 70\(^\circ\), and angle \(D\) is 110\(^\circ\)

D:

angle \(A\) is 60\(^\circ\), angle \(B\) is 50\(^\circ\), angle \(C\) is 120\(^\circ\), and angle \(D\) is 130\(^\circ\)

E:

angle \(A\) is 50\(^\circ\), angle \(B\) is 40\(^\circ\), angle \(C\) is 120\(^\circ\), and angle \(D\) is 150\(^\circ\)

Problem 2

Quadrilateral \(ABCD\) is cyclic with given angle measures.

  1. What is the measure of angle \(C\)?
  2. What is the measure of angle \(D\)?
Quadrilateral A B C D. Angle A B C is 75 degrees and angle B A D is 135 degrees.

 

Problem 3

Lin is looking at cyclic quadrilateral \(ABCD\). She says, “I’m not convinced that opposite angles of cyclic quadrilaterals always add up to 180 degrees. For example, in this diagram, suppose we moved point \(A\) to a different spot on the circle. Angle \(BCD\) would still measure 100 degrees, but now angle \(BAD\) would have a different measure, and they wouldn’t add up to 180.”

Do you agree with Lin? Explain or show your reasoning.

Quadrilateral A B C D inscribed in a circle. Angle B A D is 80 degrees, angle B C D is 100 degrees, given angles are opposite each other.

Problem 4

Line \(AC\) is tangent to the circle centered at \(O\) with radius 3 units. The length of segment \(AC\) is 4.5 units. Find the length of segment \(AB\).

Circle with center O. Triangle ACO. AC = 4.5.
A:

\(3+\sqrt{29.25}\) units

B:

\(\sqrt{29.25}\) units

C:

\(\text-3+\sqrt{29.25}\) units

D:

26.25 units

(From Unit 7, Lesson 3.)

Problem 5

Technology required. Line \(PD\) is tangent to a circle of radius 1 inch centered at \(O\). The length of segment \(PD\) is 1.2 inches. The length of segment \(AB\) is 1.7 inches. Han is trying to figure out if \(C\) or \(B\) is closer to \(P\). He uses the Pythagorean Theorem to find the length of \(OP\). Then he subtracts 1 from the length of \(OP\) to determine how far \(C\) is from point \(P\).

  1. How far is \(B\) from point \(P\)?
  2. Which point is closest to \(P\)? Explain your reasoning.
Circle with points and lines.

 

(From Unit 7, Lesson 3.)

Problem 6

In the diagram, the measure of angle \(ACB\) is 25 degrees. What is the measure of angle \(AOB\)?

Circle, center O. Points A,  B and C lie on circle. Angles A O B and A C B.  
(From Unit 7, Lesson 2.)

Problem 7

Which statement must be true?

A:

A diameter is a chord.

B:

A chord is a radius. 

C:

A chord is a diameter. 

D:

A central angle’s vertex is on the circle.

(From Unit 7, Lesson 1.)

Problem 8

A circle and line are drawn. How many intersection points are possible? Select all possible answers. 

A:

0

B:

1

C:

2

D:

3

E:

4

(From Unit 6, Lesson 13.)