1. Unit 8
Circles
By: Abhishek and Prani

2. Definitions Circle: all points are equidistant from the center
Center: the point all other points are equidistant from
Radius: segment with endpoints at the center and on the circle
Chord: segment with endpoints on the circle
Diameter: Chord that contains the center
Tangent: line that intersects the circle only once
Secant: line that intersects the circle twice
Arc: piece of the circumference
Minor Arc: an arc less that 180°
Major Arc: an arc greater than 180°
Semicircle: an arc equal to 180°
Central Angle: angle with vertex at the center of a circle.

3. Theorems of Unit 8
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
If two tangent segments are drawn to a circle from the same external point, then those segments are congruent.
If one chord is a perpendicular bisector of a second chord, then the first chord is a diameter.
If a diameter ( or piece thereof) is perpendicular to a chord, then the diameter bisects the chord and its arc. 1) 2)
3) & 4)

4. Theorems of Unit 8 (cont.)
5. In the same circle or in congruent circles, if two chords are equidistant from the center, then they are congruent.
6. In the same circle or in congruent circles, If two chords are congruent, then they are equidistant from the center.
5) & 6)

5. Theorems of Unit 8 (cont.)8)
7. The measure of an inscribed angle is one half the measure of its intercepted arc.(Vertex on circle)
8. If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
9. A triangle can be inscribed in a circle iff it is a right triangle.
10. A quadrilateral can be inscribed in a circle iff its opposite angles are supplementary. 7) 9)
10) A B C D

6. Walk Around Problems Walk around problems with common tangents.
2. Walk around problems with combined radii.
Using the two tangent theorem, you can solve the problem by working your way around the shape. First start off with one portion as “x” and work your way around clockwise or counterclockwise solving for the other side lengths.
Using the fact that all radii of a circle are congruent you can label one of the three circles radii as “x” and then again move either clockwise or counterclockwise to solve for the side radii. D A x B x C A x B x C

7. Angle Measurement In Circles
when angle is…
inside the circle
on the circle
outside the circle
To find the measure of the angle inside the circle, if it is a central angle then the angle is the same as its arc, if it is inside then it is half of the arc across the angle plus its vertical angle’s arc.
To find the measure of an angle on the circle then you would measure half of its corresponding arc.
To find the measure of an angle outside the circle then you would take half of the difference of the two intercepted arcs. A B C
∠A= arc BC A B C D 1
∠1=½(arc DC + arc AB) A
∠C = ½ arc AB C B A B C D E
∠A = ½ (arc CD - arc BE)

8. Finding Segment Lengths in Circles
Formulas for…
2 chords
2 secants
1 secant & 1 tangent A R Q
RX × SX = QX × PX C X B P S P F E
AP² = CP × BP D G
DF × EF= HF × GF H

9. Real Life Usage
GPS satellites orbit about 11,000 miles above Earth. The mean radius of Earth is about 3959 miles. Because GPS signals cannot travel through Earth, a satellite can only transmit signals as far as points A and B, as shown. Find BA and BC to the nearest mile.

10. Example #1 Solution: Find the value of x
x = 7, you can get this answer by using the theorem that states if the diameter of a circle is perpendicular to a chord then the chord is split into half. So then you would set 4x + 5 = 6x - 9. To solve this you would first subtract 4x from both sides to get rid of the 4x on one side, then you would add 9 on both sides to get rid of the -9 on the right side of the equation. this would now leave you with 14 = 2x. TO get x by itself you would divide 2 from both sides and then you would get your answer of x equal to 7.
Find the value of x

11. Example #2 Solution: Find the length of CD
CD = 23, you would get this by, I first started by labeling the congruent tangent lines, formed from the vertex of C, as x.Then I went counterclockwise and figured out that BG is 15-x, also because of the two-tangent theorem BF is equal to BG. To figure out AF I did 10 - (15-x) and got x-5, again because of the two tangent theorem AE equals AF, so AE also equals x-5. then to figure out ED i did 18-(x-5) which equals 23-x. Because ED is equal to DH then we can tell that CD= 23-x +x, that cancels out the x’s and that leaves us with CD equaling 23. A B C D
AB= 10
BC= 15
AD= 18 G F H E

12. Example #3
Find the value of x
Solution:
x=2.3,you would use the 1 secant and 1 tangent rule to solve this problem. you would set up 16 squared equal to (x+2 +14) and multiply that by 14. if you simplify that equation it would be 256 = 14x +244, then subtract 244 from both sides and get 32= 14x. to get x by itself you would divide both sides b y14vand get x=2.3 16 14
x+2

13. Connections to Other Units
This unit is similar to unit 11. In unit 8 we learned how to find the area of circles, this helps us in unit 11 by letting us find surface area and volume of a cylinder, cone, and sphere.
SA= πr²l
V= ⅓πr²h
A=πr²
SA= 2πr² + 2πr
V= πr²h
SA= 4πr²
V=4/3πr³

14. Common Mistakes/Struggles
One common mistake is mixing up the radius of a circle with the diameter. A solution is paying attention to whether the line goes halfway or all the way across the circle.
Another common mistake is thinking that all chords bisect each other and/or are perpendicular to each other instead of just diameters or their parts. A solution is making sure the line that is bisecting actually goes through or has an endpoint at the center of the circle.
A third common mistake is switching the properties of different kinds of angles in, on, or outside the circle. A solution is paying attention to where the vertex of the angle is.

15. Hope that helped everyone, Thank you!