Section 1 Circles and Circumference

Chapter 10 – Circles

Section 1 – Circles and Circumference

I can identify central angles, inscribed angles, circumscribed angles, diameters, radii, chords, and tangents.
I can prove that all circles are similar.
I can define as the ratio of a circle’s circumference to its diameter.
I can use algebra to demonstrate that because is the ratio of a circle’s circumference to its diameter that the formula for a circle’s circumference must be C = ?d.

Radius – a segment with endpoints at the center and on the circle. (denoted by “r”)
Chord– a segment with endpoints on the circle.
Diameter – a chord that passes through the center and is made up of collinear radii. (it touches 2 sides of a circle and goes through the center) (denoted by “d”)

*Formulas – r = ½ d and d = 2r

1. (a) Name the circle and identify a radius.(b) Identify a chord and a diameter of the circle.

2. (a) If RT = 21cm, what is the length of ?

Use the figure below to answer (b) and (c)

(b) If QT = 11 meters, what is ?

3. The diameter of circle X is 22 units, the diameter of circle Y is 16 units, and WZ = 5 units. Find .

Circumference – the distance around the circle. The formula is: C = 2r

4. A series of crop circles was discovered in Alberta, Canada on September 4, 1999. The largest of the three circles had a radius of 30 feet. Find its circumference.

5. Find the diameter and radius of a circle to the nearest hundredth if the circumference of the circle is 65.4 feet.

Inscribed – a polygon is inscribed in a circle if all its vertices lie on the circle.
Circumscribed – a circle is circumscribed about a polygon if it contains all the vertices of the polygon. A polygon is circumscribed about a circle if it contains the circle (the circle touches each side of the polygon).

6. (a) Find the exact circumference of circle K.

(b) Find the exact circumference, if an inscribed right triangle with legs 7 meters and 3 meters long .

Homework – Page 687 – 690 (11 - 39 ODD, 48, 49, 50)

Section 2 – Measuring Angles and Arcs

I can identify central angles, inscribed angles, circumscribed angles, diameters, radii, chords, and tangents.
I can described the relationship between a central angle and its intercepted arc.
I can define the radian measure of an angle as the ratio of arc length to its radius, and calculate a radian measure when given an arc length and its radius.
I can convert degrees to radians using the constant of proportionality (2?x angle measure/360°).
I can use similarity to derive the formula for the area of a sector.
I can find the area of a sector.
I can use similarity to calculate the length of an arc.

Central Angle – An angle with a vertex at the center of the circle. The sides of every central angle are 2 radii of the circle. The sum of all the central angles in a circle always add to 360 degrees (ex: AEB)
Minor Arc – the shortest arc connecting 2 endpoints on a circle (less than 180 degrees). The symbol for a minor arc is m (only 2 letters)
Major Arc – the longest arc connecting 2 endpoints on a circle (greater than 180 degrees). The symbol for a major arc is m (3 letters)
Semicircle – an arc with endpoints that lie on a diameter (=180 degrees exactly). When describing a semi circle, use 3 letters (ex m)

*ALL arcs are congruent to their corresponding central angles!! (this is important to remember)

*If converting a percent of a circle into a measurement. Take the decimal form of the percent (ex: 36% would be 0.36) and multiply it by 360.

1. Find the value of x:

2. is a radius of circle C. Identify each arc as a major arc, minor arc, or semicircle. Then find its measure.

(a)m(b) m(c) m

3. Refer to the circle graph.

4. Find each measure in circle M.

Arc Length – the distance between the endpoints along an arc measured in linear units. Arc length = (degree measure of arc/360) times circumference of the circle.

Arc = (x/360)(2r)

5. Find the length of . Round to the nearest hundredth.

(a)(b)

(c)

Homework – Page 696-698 (13 - 47ODD) (24)

Section 3 – Arcs and Chords

I can identify central angles, inscribed angles, circumscribed angles, diameters, radii, chords, and tangents.

*In the same circle or in congruent circles, 2 minor arcs are congruent if and only if their corresponding chords are congruent.

*If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc.

*In the same circle or in congruent circles, 2 chords are congruent if and only if they are equidistant from the center.

1. A circular piece of jade is hung from a chain by two wires wrapped around the stone.

2. In the figure, circle A = circle B and WX = YZ. Find WX.

3. In circle G, m = 150. Find m.

4. In the ceramic stepping stone below, diameter AB 5.

is 18 inches long, and chord EF is 8 inches long. Find CD.

Homework – Page 705 – 706 (7-23ODD, 31-33, 35)

Section 4 – Inscribed Angles

I can identify central angles, inscribed angles, circumscribed angles, diameters, radii, chords, and tangents.
I can describe the relationship between an inscribed angle and its intercepted arc.
I can describe the relationship between a circumscribed angle and its intercepted arc.
I can verify that inscribed angles on a diameter are right angles.
I can construct the inscribed circle whose center is the point of intersection of the angle bisectors (incenter).
I can prove that the opposite angles in an inscribed quadrilateral are supplementary.
I can construct the circumscribed circle whose center is the point of intersection of the perpendicular bisectors (circumcenter).

Inscribed Angle – an angle that has a vertex on the side of a circle and the sides of the angle are chords of the circle. The arc that corresponds to the inscribed angle is called the intercepted arc. ()
Intercepted Arc – corresponds with the inscribed angle and has points on the circle. Lies on the interior of the inscribed angle. ()

Formula: Inscribed angle = ½ intercepted arc

*If 2 inscribed angles have the same intercepted arc or congruent arcs, then the inscribed angles are congruent.

*If the inscribed angle of a triangle intercepts the diameter or a circle, then the inscribed angle =90.

*If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary (add to 180)

1. Find each measure:

Guided Practice

2. Find mR

3. Write a two-column proof:

Given:

Prove:

4. Find the value of x and mB.

4. If mA = 7x + 2 and mB = 17x – 8, find x.

5. An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mS and m T.

Homework – Page 713 – 714 (11-20 and 23-35) omit #31

Section 5 – Tangents

I can verify that the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
I can verify that inscribed angles on a diameter are right angles.
I can construct a tangent line from a point outside a given circle to the circle.

Tangent– a line on the same plane as a circle and touches the circle at 1 point only(called the point of tangency), but does not go inside the circle at any point.
Common Tangent – a tangent line that is tangent to 2 circles on the same plane at the same time.

*A line is tangent to a circle if and only if it is perpendicular to a radius drawn at the point of tangency.

*If 2 lines are tangent to the same circle and meet outside the circle, then those 2 lines are congruent.

1. KL is a radius of circle K. 2. Determine whether GH is tangent

Determine if KM is tangent to circle K. to circle F.

Justify your answer.

3. (a) In the figure, WE is tangent to circle D at W. Find the value of x:

4. AB and BC are tangent to circle Z. Find the value of x:

5. The round cookies are marketed in the triangular package to pique the consumer’s interest. If triangle QRS is circumscribed about circle T, find the perimeter of triangle QRSHomework – Page 722-723 (13-27, 29, 35)

Section 6 – Secants, Tangents and Angle Measures

I can describe the relationship between two secants, a secant and a tangent or two tangents in relation to the intercepted circle.

Secant – a line that intersects a circle in exactly 2 points. It goes completely through the circle.
Tangent– a line that intersects or touches the circle in only 1 spot.

*3 types of problem:

(1) 1 secant and 1 tangent meeting at the point of tangency on the circle

(formula – angle = ½ intercepted arc)

(2) 2 secants meeting inside the circle

(formula – angle = ½ sum of intercepted arc + vertical angles intercepted arc)

(3) 2 secants, 2 tangents, or 1secant and 1 tangent meeting outside the circle

(formula – angle = ½ the difference of the larger intercepted arc minus the smaller intercepted

arc)

*These formulas and pictures of these are on page 731 – very good descriptions

1. Find x:

2. Find each measure

3. Find each measure

4. PHYSICS. The diagram shows the path of a light ray as it hits a cut diamond. The ray is bent, or refracted, at points A, B, and C. If m = 96 and m = 35, what is m

Homework – Page 732-733 (8-29)

Section 7 – Special Segments in a Circle

I can describe the relationship between two secants, a secant and a tangent or two tangents in relation to the intercepted circle.
I can find the lengths of segments formed by lines that intersect circles.

*3 scenarios for solving segment problems and how to solve:

(1) 2 chords that meet inside a triangle (the products of the segments of one chord = the products of the segments of the second chord)

(2) 2 secants meeting outside the circle (the product of the whole secant times just the outside segment of that secant = the product of the whole other secant times just the outside of its outside portion.

(3) 1 secant and 1 tangent meeting outside the circle (the square of the tangent = the whole secant times just the outside section of that secant)

1. Find x:

2. Biologists often examine organisms under microscopes. The circle represents the field of view under the microscope with a diameter of 2mm. Determine the length of the organism if it is located 0.25mm from the bottom of the field of view. Round to the nearest hundredth.

3. Find x:4. Find x if LM is tangent to the circle

Homework – Page 740 (6-21, 26)

10.8 – Equations of a Circle

I can use the Pythagorean Theorem to derive the equation of a circle, given the center and radius.
I can complete the square to find the center and radius of a circle when given an equation of a circle.

IMPORTANT – ON THE ACT!!!

(x – h)2 + (y – k)2 = r2

Center (h, k) and radius = r

1. Write the equation of each circle:

(a) center at (3, -3) and radius = 6

(b) the circle graphed below:

2. Write the equation of each circle:

(a) center at (-3, -2) and passes through the point (1, -2)

(b) the circle graphed below:

3. The equation of a circle is (x – 2)2 + (y + 3)2 = 4. State the center and the radius. Then graph.

4. Find the equation of a circle with the endpoints (-2, 10) and (8, 6)

**Writing the Equation of a Circle in standard form by COMPLETING THE SQUARE!!!!

Example:

4x2 + 4y2 – 16x – 24y + 51 = 0

Step 1: Move the constant number to the other side  4x2 + 4y2 – 16x – 24y = -51

Step 2: Group the “x’s” and “y’s” together  4x2 – 16x + 4y2 – 24y = -51

Step 3: What is multiplied on the “squared” terms (it will always be the same number with circles), divide

EVERY term in the equation  x2 – 4x + y2 – 6y = -51/4

Step 4: Group the “x’s” and the “y’s”, then complete the square by dividing the x and y term coefficient by 2

and squaring it and adding this to the end of the grouping. ALSO add these numbers to the other

side of the equation  (x2 – 4x + 4) + (y2 – 6y + 9) = -51/4 + 4 + 9

Step 5: Factor each perfect square trinomial on the left, and add the numbers on the right

 (x – 2)2 + (y – 3)2 = ¼

Step 6: Find the center and radius of the circle Center (2, 3) and Radius = ½

5 – 7: Find the center and radius of each circle:

5. x2 + y2 – 2x – 4y – 4 = 0

6. -4x + 2y2 = -8y – 2x2 - 8

7. 100x2 + 100 y2 – 100x + 240y – 56 = 0

Homework – Page 747-750 (11-33 ODD, 20, 28, 34, 37) omit 25 and Worksheet problems