Name: ___________________________

Geometry Period _______

Unit 13: Circle Theorems - Part 2

In this unit you must bring the following materials with you to class every day:

Calculator

A Pencil This Booklet

Colored pencils/pens/highlighters

A device

Headphones!

Please note:

You may have random material checks in class

Some days you will have additional handouts to support your understanding of

the learning goals in that lesson. Keep these in a folder and bring to class every

day.

All homework for part one of this unit is in this booklet.

Answer keys will be posted as usual for each daily lesson on our website

Lengths of Segments in Circles Today’s Goal: How do we find the measure of segment lengths formed by intersecting chords? Turn and Talk! - Answer the following questions with your neighbor!

1) Given that m ACD m ABD, can you conclude that ΔCEA ~ ΔBED?

(Recall, we need 2 congruent angles for triangles to be similar)

2) Based on your answer above, what do we know about the corresponding sides in these two triangles?

Corresponding sides of _________________ triangles are in _________________________________

3) Fill in the proportion of Corresponding sides!

*remember you can use the triangle’s initials to help you!

4) What do we know about the cross product of proportions? Find the cross products of the proportion above.

______________________ of proportions are __________________

5) What do you notice about and ? How about and ?

Rule: To solve for length involving intersecting chords:

13-6 Notes

SPECIAL ADDITION TO THE RULE:

If a radius (or diameter) is ___________________________ to a

chord, then it also bisects that chord

Let's Try It!

1) Chords and intersect inside the circle. What is the length of ?

2) Chords and intersect at E with AE = 9 and EB = 1. a) Sketch it! b) If CD = 6 and CE = x, what expression represents ED in terms of x? c) Find CE. d) Find ED.

e) If CD AB what special segment would AB be in the circle?

PRACTICE

1) Chords and intersect at E. If AE = 6, EB = 10, and ED = 12, find CE.

Let’s draw it first!

2) a) Chords AB and CD intersect at E. If AB = 20, AE = 8 and CE = 6, find ED

b) If 200. What is the measure of <AED?

3) Chords AB and CD intersect at E. If CE = 4, ED = 12, and EB is 2 more than AE, find AE and EB.

4) In circle O, diameter DB is perpendicular to chord AC at E.

If DB = 34, AC = 30, and DE > BE, what is the length of BE?

(a) 8 (c) 9

(b) 16 (d) 25

5) Given tangent at D, secant through center, AB || DC, and m . Find the measures of angles 1-7.

13-6 Homework Directions: Answer the following questions to the best of your ability.

1) Find the value of x.

2) In circle , diameter is perpendicular to chord at . If and , what is ?

3) In the diagram of circle O below, chord intersects chord at E, , , , and .

Solve for AE.

1) 32

2) 16

3) 10

4) 8

3) In the diagram below of circle O, diameter AB is perpendicular to chord CD at E. IF AO = 10 and BE = 4, find the

length of CE.

5) In the accompanying diagram of circle O, chords AB and CD intersect at E. If AE = 3 , EB = 4, CE = x and ED = x-4, what

is the value of x?

What is m𝐴𝐵 ? What is the m𝐴𝐶 ?

What is m𝐴𝐷? What is m𝐴𝐸?

What do you notice? Is there any relationship shared between all the lengths that are given and the one’s you found?

Learning goal: What are the relationships between secant and tangent lengths outside of circles?

How do we find measures of secant and tangent lengths?

Consider the following diagram with marked lengths.

When solving for segments involving secants/tangents:

Rule:

13-7 Notes

Let’s try some! Example 1: AF is tangent to circle at F and secant and secant ABC intersects the circle at B and C. If AC = 8, and AB = 4.

Find AF to the nearest 10th.

Example 2: Secants SQP and TRP intersect at P.

If PT = 18, PR = 4 and SQ = 1, find PQ.

Finding Lengths Given Two Tangents

Connect Points A and C on the circle below.

1) What types of angles are ?

2) What is the relationship between these angles?

3) What type of triangle is ? how do you know?

4) What must be true about the length of ?

Theorem:

If two ____________________ meet at an exterior point, then those tangents are ___________________.

Example 3: In the accompanying diagram, circle O is inscribed in ABC so that the circle is tangent to at F, to at

E, and to at D. If AF = FB = 5 and DC = 7, find the perimeter of ABC.

Congruent Chord Theorem

We learned that When two chords are congruent, the arcs “outside of them”, or

subtended by them are congruent.

Think about it!

What do you think is true about the distance of chords AB and CD from

the center?

Theorem: Chords equidistant from the center of the circle are _______________

Example 4) Given circle O, find x. We know…

*the congruent pieces marked are _____ to chords

*the chords are ________________

PRACTICE

1) Secants SQP and TRP intersect at P. If PT=15, PR = 6 and PS = 10, find PQ.

2) In the diagram, PA is tangent to circle O and PBC is a secant. If PA = 4 and BC = 6, find PB.

3) In the accompanying diagram, cabins B and G are located on the shore of a circular lake, and cabin L is located near

the lake. Point D is a dock on the lake shore and is collinear with cabins B and L. The road between cabins G and L is 8

miles long and is tangent to the lake. The path between L and dock D is 4 miles long.

What is the length, in miles, of ?

(1) 24 (2) 8 (3) 12 (4) 4

4)

P

6) Given: Circle O, AB = CD. Find x.

7) Putting it all together! Use the following diagram to answer the questions below. PA is tangent to circle O at A, and

secant PBC intersects circle O at B and C.

a) If m = 150 and m = 110, find m P.

b) IF PB = 15 and PC = 60, fnd PA.

c) IF PB = 4 and BC = 5, find PA.

d) IF PA = 5 and PB = 2, find PC.

e) If PA = 8 and BC = 12, find

i. PB

ii. PC

8) Given: Circle O, CD = 16, AB = 16, OB = 10. Find OF.

9) Given circle with two tangents and a secant. Find x.

13-7 Homework

1. READ & HIGHLIGHT:

*NEW TERM* Common Tangents: the lines that you can draw that are tangent to both circles simultaneously

There are 2 cases you need to be aware of:

Case 1:When circles are separate from each other, there are 4 possible tangent lines that can be sketched in:

Case 2:When circles are tangent to each other, there are 3 possible tangent lines that can be sketched in:

Sample Question on topic above:

In the diagram below, circle A and circle B are shown. What is the total number of lines of tangency that are common

to circle A and circle B?

2. In the accompanying diagram, segments , , and are tangent to circle O at A, B, and C, respectively. If

, , and , what is the measure of ?

3) In the diagram below of circle O, secant intersects circle O at D, secant intersects circle O at E, ,

, and DB=10.

What is the length of ?

1) 4.5

2) 7

3) 1

4) 14

4) In the accompanying diagram, secant intersects circle O at D, secant intersects circle O at E, ,

, and AB=21. Find AD.

5) In the accompanying diagram, is tangent to circle O and is a secant. If and , find PB.

6) Given: circle with secants and chords such that LR>RT and: LM = 7 MN = 8 TN = 10 PR = 3 RM = 6 LT = 11

Find:

i. PT

ii. PN

iii. LR (HINT: look at givens!))

Today’s Learning Goal: How do you solve big circle problems?

What are big circles problems?

A "BIG" circle refers to a question that requires the use of all (or most) of your circle angle formulas in

one problem. It may also be necessary to apply other strategies to find missing angles.

Tips for Solving Big Circles!

1) Always solve for arcs first

2) Solve the parts in the order they are presented in for the problem

3) GO SLOW

4) Highlight parts you're solving for (in different colors)

5) Fill in/mark up the diagram as you go (It will probably be used again later)!

In your teams: Feel free to skip around and try to answer AS MANY QUESTIONS AS YOU CAN in this class period. Check in AFTER EACH

Quesitons. I will be recording which problems your teams solves correctly.

1. If PC is a secant , PA is a tangent in circle O, and , Find:

For a-d, record answer here AND in diagram!

a)

b) <ACP

c) <PAC

d) <APC

13-8 Notes

2. In the circle shown below, the measure of , and the measure of <RPQ = 50o. What is the measure of ?

3. If Triangle ABC is inscribed in circle O, <AEB = 55o and , Find:

Diagram not drawn to scale.

What circle angle theorems are you noticing coming up a lot? STATE THEM HERE:

CHECK IN WITH YOUR TEACHER TO MAKE SURE YOU ANSWERED 1-3 CORRECTLY! Keep GOING!

a) 𝐴��

b) m<ADB

c) m<EAC

d)m<ACB

4. In the accompanying diagram of circle O, PBA and PCD are secants, chords AC and BD Intersect at E, BA CD, chord

BC is drawn, <ABD = 65o and ,

Find:

a) Find and label the measures of ALL missing arcs. ADD ALL ARC MEASURES IN DIAGRAM!

b) m<ACD

c) m<P

d) m<DBC

e) m<ACP

5. Given circle O, AB is a diameter, DA and DC are tangents and:

a) FIND AND FILL INTO DIAGRAM ALL ARC MEAURES!!

b) Find:

6. In the accompanying diagram of circle 0, AE and FD are chords, AOBG is a diameter and is extended to point C. CDE

is a secant, AE||FD and : : = 3:2:1. In addition AC = 24, GC =4 EC =48 AO=10 OB=2 FB= 16.

Solve for:

1.

2.

3.

In addition AC = 24, GC =4 EC =48 AO=10 OB=2 FB= 16.

4. DC

5. DB

13-8 Homework

1. Chords and intersect at E in circle O, as shown in the diagram below. Secant and tangent are

drawn to circle O from external point F and chord is drawn. The , ,

(a) Determine .

(b) Find m

(c) Find m CDF

(d) Find m< AFB

2. Given circle O with diameter AB, determine and state the value of X. SHOW ALL WORK in working space and in

diagram!

3. Given: AD CB MARK IT!

Explain why

4. Given MARK IT!

Explain why

5. Consider the following diagram of Circle E with congruent arcs AB and CD.

What short cut method would you use to prove the two triangles are congruent?

State the congruent parts that will justify your response.

Today’s Goal: How can theorems of circles, angles and segment lengths are used in formal proofs?

Thus far in the unit we have discussed relationships between angles and segments, how they are formed and their length

and measure. Let’s use the same relationships and see how we can apply them to proofs!

Class Discussion Algebraic Solving Geometric Reasoning

1) Given circle O- what would be the measure of ?

Which two angles MUST be congruent?

1) Same relationship-but in a proof scenario! (no angle measures!)

Which two pairs of angles MUST be congruent? What’s the reason? Find this Proof piece in your packets, what is the name of this theorem?

2) Given circle O- what would be the measure of ?

Which two arcs must be congruent?

2) Same relationship-but in a proof scenario! (No angle measures!) Which two angles MUST be congruent? What’s the reason? Find this Proof piece in your packets, what is the name of this theorem?

13-9 Notes

3) Given circle O- what would be the measure of ?

Which two chords must be congruent?

3) Same relationship-but in a proof scenario! ( no angle measures!) Which two chords MUST be congruent? What’s the reason? Find this Proof piece in your packets, what is the name of this theorem?

PARTNER RE-ACTIVATION! Using these proof pieces, we can now explain why certain angles and segments must be congruent without

knowing their measure.

What types of proofs require that we show congruent sides and/or angles?

Can you recall the “short-cuts” we’ve used for these types of proofs?

Let’s try one together! Circle Proof Example

1) Given: AC || DB

Prove:

Practice in Your Teams!

Warm-Up:

Using the Given information, state the relationship each triangle has (Similar or Congruent)

Work out:

For the following problems, look at each diagram and state which proof piece(s) you can use based on the givens and any statement that it supports. The first one is done for you. Make sure you explain WHY the statement is true!

MODEL EXAMPLE 1) Given :Diagram of circle O. What are congruent angles?

<A <B by Inscribed Angle Theorem

2)

What shortcut method supports that? What shortcut method supports that?

(You may need to identify another congruent part!)

Thus, what else is congruent?

What proof piece supports this?

6) Given: Circle O, with Chords AC, CB, and Diameter AOB, radius OC, Tangent CP and Secant AOBP.

Prove: PCO is a right triangle

6) Ready? Try a full proof now! Make sure you work together, and use your proof pieces to help you get ideas on where

the statements. The rubric is given to you to make sure everything is included!

Statement Reason

1. Circle O, with Chords AC, CB, and Diameter AOB, radius OC, Tangent CP and Secant AOBP

1.

2. PC CO 2.

3. 3. Definition Perpendicular

4. 4.

Statement Reason

Statement Reason

1. 1.

2. CAE DBE 2.

3. 3.

4. AEC BEC 4.

5. 5.

6. 6.

a)

b)

c)

d) If chords are

parallel, arcs in

between are

congruent!

How can

inscribed angle

theorem help

you?

13-9 Homework

Directions: Answer the following questions to the best of your ability. Show all work.

1. In the diagram below, is inscribed in circle P. The distances from the center of circle P to each side of the

triangle are shown.

Which statement about the sides of the triangle is true?

1)

2)

3)

4)

2. Kimi wants to determine the radius of a circular pool without getting wet. She is located at point K, which is 4 feet

from the pool and 12 feet from the point of tangency, as shown in the accompanying diagram.

What is the radius of the pool?

1)

2)

3)

4)

3. Line segment is tangent to circle O at A. Draw it! Which type of triangle is always formed when points A, B, and

O are connected?

1) right

2) obtuse

3) scalene

4) isosceles

4. In the accompanying diagram, , , and is the diameter of circle O. Write

a proof that shows and are congruent.

5. Given that Chords and intersect at E, and . Prove that .

6. Given:

Prove:

Statement Reasons

1. 1.

2. 2.

3. 3.

4. 4. Congruent Chords Theorem

5. BD BD 5.

6. 6.

Statement Reason

1. 1.

2. 2. Congruent Chords Theorem

3. OX OY 3.

4. 4.

5. XB YB 5.

6. 6. SSS