ATM - Circles Lesson

Circles

Circles are two-dimensional objects with an outer ring defining the shape, the circumference. Before we start in on the rules of circles, try the matching vocabulary activity.

two circles labeled with various point A - J

The length of the circumference of the circle is C = 2πr = πd , the distance around the circle. The radius r is half the diameter or the diameter d = 2r.

The area of a circle is found by using the formula A = πr².

There are 360^º in a circle and 180^º in a semi-circle.

Tangents to a Circle

1) Two rays tangent to a circle from an exterior point create congruent segments.

Segment NP is congruent to segment NM so their lengths are equal.

2) Segment CP is congruent to segment CM.

Both of the segments are a radius of the circle so they must be the same length.

3) A radius drawn to the point of tangency is perpendicular to the tangent line.

Radius CP is perpendicular to segment NP meaning that there is a 90-degree angle at the intersection.

4) The measure of exterior angle MNP is half the measure of the farther arc (major arc MP) minus the nearer arc (minor arc MP).

If major arc MP has a measure of 200 degrees and minor arc MP is 160 degrees, then $LaTeX: \angle MNP=\frac{1}{2}\left(200-160\right)=\frac{1}{2}\left(40\right)=20^\circ$ $\angle MNP=\frac{1}{2}\left(200-160\right)=\frac{1}{2}\left(40\right)=20^\circ$

Example 1: If segment MN = 2x + 5 and segment NP = 3x - 1, find the value of x and then the length of the segments in the picture above.

Since the two segments are tangent to the circle from an exterior point they are congruent; the same length.

Solve 2x + 5 = 3x - 1

6 = x Combine like terms.

The length of the segment MN is 2(6) + 5 = 12 + 5 = 17.

The length of the segment NP is 3(6) - 1 = 18 - 1 = 17.

In the video below, right triangles are introduced into the circle with the tangent rules above.

Chords and Major and Minor Arcs

1) A chord touches the circle at two points.

a) Diameter, AE, is a special chord that goes through the center of the circle.

b) Segment FH is a chord.

2) Radius CG is perpendicular to chord FH. Since the radius is perpendicular to the chord, radius CG bisects chord FH dividing it into two equal parts, segment FI is congruent to segment IH.

3) Major arcs have a measure greater than 180 degrees, a minor arc is less than 180 degrees. A semi-circle is 180 degrees.

a. Arc ABE = AE is 180 degrees. Indicating the B in the middle states the direction to go around the circle.

b. Arc DB is a minor arc as the measure will be less than 180 degrees.

c. Arc AED is a major arc as the measure will be greater than 180 degrees, arc AE + arc ED.

Watch the video for further explanation on chords and arcs.

Angles and Arc Measure

Definitions and formulas for finding the angle measure are needed with this diagram.

1) Central Angle examples are <ACG, <GCE, <ECD, <DCB, <BCA, <ACD, <ACE. For each of these, the measure of the central angle is the measure of the arc the angle intercepts.

2) Arc measure is the same as the measure of the central angle.

a. <ACE is the diameter so since this angle is half a circle, the angle measures 180 degrees. So arc ABE is also 180 degrees.

b. If <ACG is 60 degrees then the measure of arc AG is 60 degrees since <ACG is the central angle.

3) Angle AFG and angle EAF are inscribed angles because the vertex of the angles lies on the circle.

a. The measure of the inscribed angle is half the measure of the intercepted arc. So if the measure of arc AG is 60 degrees, then the measure of <AFG is 30 degrees.

m<AFG = (1/2)(60) = 30 degrees

b. If the measure of angle EAF is 10 degrees, what is the measure of the arc EF? Work this backward by writing the rule, filling in what you know, and then solving for the missing variable.

m<EAF = (1/2)(m arc EF) Write the equation.

10 = (1/2)(m arc EF) Substitute.

20 = measure of arc EF Isolate the missing part.

Radian Measure

Radians are another way to measure angles in a circle. One radian is the measure of the central angle subtended by an arc that is equal to the radius of the circle.

If there are 360 degrees in a circle, then there are 2π radians in a circle. Another way to think about that ratio is 180 degrees to π radians.

1) To change degree measure to radian measure (measure in π):

60 degree * π/180 degrees = π/3

2) To change radian measure to degree measure:

π/3 * 180/π = 60 degrees

View the video below to learn more about angle measure and arc measure.

Segment Measure and Interior Angle Measure

Vertical Angles: Angle YVZ and angle WVX are vertical angles and their measure will be congruent. Angle WVZ and angle XVY are vertical angles and their measure will be congruent.

Angles Inside the Circle: Angles inside the circle will be equal to half the sum of their intercepted arcs (on both sides of the angle)

The measure of $LaTeX: \angle WVY=\angle YVZ=\frac{1}{2}\left(arcWX+arcYZ\right)$ $\angle WVY=\angle YVZ=\frac{1}{2}\left(arcWX+arcYZ\right)$

The measure of $LaTeX: \angle WVZ=\angle YVX=\frac{1}{2}\left(arcWZ+arcYX\right)$ $\angle WVZ=\angle YVX=\frac{1}{2}\left(arcWZ+arcYX\right)$

Segments Inside the Circle: To solve for the measure of the segments WV, VY and XV, VZ set the product of their partial lengths equal.

WV * VY = XV * VZ

Example 2: If the measure of arc WX = 25 and the measure of arc YZ = 50, then the measure of $LaTeX: \angle YVZ=\frac{1}{2}\left(25+50\right)=\frac{1}{2}\left(75\right)=37.5$ $\angle YVZ=\frac{1}{2}\left(25+50\right)=\frac{1}{2}\left(75\right)=37.5$

Example 3: If the measure of segment WV = 2x + 5, VY = 3, XV = 4x - 2 and VZ = 2, find the value of x and then the measures of each of the chords.

WV * VY = XV * VZ Rule.

(2x + 5) * 3 = (4x - 2) * 2 Substitute.

6x + 15 = 8x - 4 Distribute.

19 = 2x Combine Like Terms

19/2 = x or x = 9.5

WV = 2x + 5 = 2(19/2) + 5 = 19 + 5 = 24

WY = WV + VY = 24 + 3 = 27 is the chord length

XV = 4(19/2) - 2 = 2(19) - 2 = 38 - 2 = 36

XZ = XV + VZ = 36 + 2 = 38 is the chord length

Secants, Tangents and Segment Length

Secants touch the circle at two places with part of the lines outside the circle. Tangents touch the circle in one place and never go inside the circle. To find the segment lengths of either of these situations above, use the following rule:

part * whole = part * whole

The part is the outside length and the whole is the total length of the segment from the exterior point until the segment last touches the circle.

Example 4: If AB = x, BC = 6, AE = 5 and ED = 4, find the length of each of the secant lines.

part * whole = part * whole

(x) * (x + 4) = 5 * (5 + 4)

x² + 4x = 45

x² + 4x - 45 = 0 Set up to find the solutions (zeros)

(x + 9)(x- 5) = 0 Factor

x = -9 and x = 5 Zeros. These must be checked.

A length of -9 is not appropriate for x as length is positive.

The answer is x = 5.

The only difference between the secant and tangent is that the part and whole for the tangent are the same value.

Sector Area and Arc Length

A sector of a circle is a portion of the area. A sector that is a semi-circle has half the area of a circle.

Find the area of a sector by using a proportion between area and degrees.

$LaTeX: \frac{area\:of\:\sec tor}{area\:of\:circle}=\frac{measure\:of\:the\:\sec tor\:arc}{360^\circ}$ $\frac{area\:of\:\sec tor}{area\:of\:circle}=\frac{measure\:of\:the\:\sec tor\:arc}{360^\circ}$

Example 5: If the radius of the circle is 5 feet, m<C is 120 degrees find the area of the sector.

Note that m<C is a central angle, so the measure of the arc = 120 degrees.

$LaTeX: \frac{\sec tor\:area}{\pi\left(5\right)^2}=\frac{120}{360}$ $\frac{\sec tor\:area}{\pi\left(5\right)^2}=\frac{120}{360}$ Substitute into the formula.

Sector area = $LaTeX: \frac{1}{3}\cdot25\pi$ $\frac{1}{3}\cdot25\pi$ Isolate variable.

Sector area = $LaTeX: \frac{25\pi}{3}$ $\frac{25\pi}{3}$ feet Solve for the sector area.

Arc length is the walking distance around the circle from point A to point B. The circumference is the walking distance completely around the circle and is represented by C = 2πr. To find the arc length of a sector, compare the sector arc length to the arc length of the circle as a proportion with the degrees of the sector arc and the degrees in a circle.

$LaTeX: \frac{arc\:length}{circumference}=\frac{arc\:measure}{360}$ $\frac{arc\:length}{circumference}=\frac{arc\:measure}{360}$

Example 6: If the radius of the circle Is 4 inches and the measure of the arc of the sector is 50 degrees, find the arc length.

The following video

The Intersection of a Circle and a Line

The equation of a circle is of the form (x - h)² + (y - k)² = r² where (h, k) is the center of the circle and r is the radius of the circle.

Example 7: Given the equation (x - 4)² + (y + 3)² = 25 , what is the center and radius of the circle.

The center of the circle is at (4,-3) and the radius is 5.

x - 4 = 0 y + 3 = 0 r²= 25

x = 4 y = -3 r = 5 (positive length)

The following video will show the process of finding the intersection of a line and a circle.

Answer the questions. For answers and detailed solutions, download this handout Links to an external site..

Review

This module has a lot of formulas with which you can apply the basic information from the beginning. Note the last problem that I did in the video was a circle line intersection. Apply what you know across all of the situations presented to you. Interconnect the ideas. The SAT Math Formula Sheet will be given to you on the exam, but the rest of the information will be used based on interpretation in problems that are asked.

Know that you will need to reason through some problems. Write down what you know, then select a formula or method to work on the problem.

Be comfortable using your calculator on the Trigonometry section, but be aware, the Special Right Triangle information will be in the non-calculator on the SAT.

Review all formulas and know how to use them, not just the formulas on the SAT Math Formula Sheet.

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