Unit 9: Circles

Radian equation?

(Theta) = arc length/radius

Area of a sector equation

.2(pi) is for radian measures, 360 is for degrees

Radian

1 radian is the angle measure needed to travel 1 radius distance around the circle

Note

2(pi) radians = 360, and (pi) radians = 180

Tangent

A line in the plane of a circle that intersects the circle in exactly one point. (Purple)

Determining tangent lines

A line is tangent to a circle if it: Intersects at only 1 point And is perpendicular to its radius/diameter

Secant

A line that intersects a circle in two points

Sector

A part of the interior of a circle between two radii (radius x 2). Two sectors have to have congruent central angles to be similar

Arc

A portion of the circumference of a circle between two radii. Two of these both need congruent central angles to be similar

Using Pythagorean theorem to determine a right triangle

A^2 + B^2 = C^2 If the sum of A and B equals C, then it is a right triangle,

Note

All circles are similar

Note about quadrilaterals

All interior angles of a quadrilateral add up to 360

Subtended angle

An angle around the center point used to find arc length. It's presumed that the subtended angle of a circle's circumference is 360

Major arc

An arc of a circle whose measure is greater than 180 degrees, represented with three letters. (Yellow)

Inscribed angles

Angle with vertex on the circle.

Finding arc measure (example two)

Another logic, just a bit more complicated: This one, we only know 93 and 38, and want arc AB (another minor). Here we know that line AD and CE are diameters, meaning the angle on both of those is 180 (but we don't actually need that bit.). Both of those lines make a vertical angle for 93, meaning 38 + that blank bit we need makes another 93. So, it's just subtraction (93 - 38 = 55) and the measure we get is 55

Finding the fraction of a circumference (process I guess)

Arc length can be defined as its subtended angle times radius (theta * r). Circumference is defined as 2piR. So to find the fraction if the circumference, we use arc length/circumference, which looks like thetaR/2piR. The radii cancel out, making theta/2pi If theta is a fraction, you can multiply it by 1/2pi.

Subtended angle from arc length and radius

Arc length/radius makes the angle (in radians)

Subtended angle from arc length

Arc length/rest of circumference = center angle/360

Find the area of a fraction of a circle

Assuming you know the fraction and radius. Take the area of the circle, and then multiply by the fraction. Example (r = 6 and 3/4 of the circle) Area = (pi)R^2 Pi * 6^2 36(pi) 3/4 * 36(pi) 108/4 * (pi) 27(pi) A = 27(pi)

Finding the arc length

Assuming you know the radius and fraction of the circle, this is the process: find circumference, then multiply that by the fraction of the circle. Example (with a radius of 7 and fraction of 1/4) Circumference = 2piR 2pi7 7*2pi 14pi 1/4 * 14pi 14/4 * pi 3.5pi

Converting radians to degrees

Basically cross multiply whatever fraction in radians with 180/pi. Example on the side

Ratio for circles

In triangles it was opposite/hypotenuse, but for circles it's arc length/radius length

Inscribed angle theorum

Inscribed angles are always half of the central angle of the same arc

Solving proportions

Looks like 11/n = 8/5 or something similar. Cross multiply and then divide out to put the variable alone. If the variable is the numerator, multiply the bottom number on both sides.

Tip

Make sure that you calculate circumference when looking at a radius.

Finding diameter with inscribed shapes (example)

Ok, inscribed angles are always half of the central angle, right? With that diameter there, it creates a central angle of 180. Since half of 180 is 90, we now know that C is 90 degrees. (inscribed angle is C) With C now being defined as a right angle, we now know that triangle ABC is a right triangle and we can use the Pythagorean theorum to find the missing side, which happens to be 17.

(Inscribed shapes) angle subtended by diameter

Okay so we need angle ISE, which is that corner there. We know that all interior angles of a triangle add up to 180, so all we need is to find another angle to go off of. Angle LIS is complimentary to another angle we need, angle SIE. We know that complimentary angles must add up to 180, so it's just 180 - 61, which is 119. Now that we have 119 along with 27, we can solve for the missing angle. 119 + 27 + X = 180 146 + X = 180 X = 34 Our missing angle is 34

Solving inscribed quadrilaterals (example)

Okay, we need to find angle WDL, but we know WIL (which is 45) and ILD (which is 109). Well, do we have any inscribed angles we can use to find WDL? The easiest one is to just draw WDL's arc, which is VERY big, it's almost the entire circle! But not all of it. Conveniently, there's a space where WIL has endpoints we can claim as chords (W and L) to make it inscribed. We know inscribed angles are half of the arc measure, and since we know WIL is 45, we also know that arc is 90, leaving the rest of the circle as what we need to find as 1 unit. Well, if a circle is 360, minus the 90 we know belongs to WIL, that would mean WDL's arc is 270. Half of 270 is 135. Therefore, WDL is 135 degrees

Note

Pi/180 radians = 1 degree

(Inscribed shapes) finding inscribed angle

So we need to find angle DEG, but we only have 1 known angle: CFG with 50 in it. Even though center point O isn't involved, we can still make inscribed angles. DEG can make an arc between C and D, making it inscribed. And you know who else is connected to those points? CFG is, and we can make it another inscribed angle. We know that two inscribed angles of the same arc are basically the same arc, making both CFG and DEG 50 degrees.

Getting arc length from subtended angle and radius

Subtended angle * radius = arc length (might need to be in radians to work)

Arc measure

The angle created by two lines from the endpoints of an arc that cross with the center. (see picture, the measure is 90)

Note about ratio

The ratio for a full circle is 2(pi)

Finding arc measure (example one)

This is all logic but here's an example if you forgot: AC is a minor arc, bc it only has two letters. This means we want the shortest distance between those points. So, the measure of AC is the full angle between those lines. Since we know two angles in that area, 70 and 104, we add them together to make 174. The measure of arc AC is 174

Finding arc measure (example 3)

This time we want a major arc, buts it's actually easier: This time we only have 69 it seems, but we actually know another. Because AB is a diameter, the whole angle in that half of the board is 180, which is what we want, the whole distance from a to b is 180, plus that extra 69 from b to c is 249. Our measure is 249. (No we couldn't have just subtracted 69 from 360 bc that makes 291, which isn't right)

Finding angles in an isosceles

Two sides are the same number and the base is different, and the angles act similarly. Two of the angles are the same with one outlier (see photo). Therefore one angle is a duplicate. Of course, all angles have to add up to 180.

Finding arc measure w/ equations (example 2)

We need to find BC, which consists of both equations 4y + 6 and 7y - 7. We can also write that as 4y + 6 + 7y - 7, which makes 11y - 1. We can use the extra 20y - 11 to complete the full circle around, which would equal 360, and that looks like this: 11y - 1 + 20y - 11 = 360 31y - 12 = 360 31y = 372 Y = 12 So, now that we know what Y is, let's solve 11y - 1 11(12) - 1 132 - 1 So the arc is 131 degrees

Finding arc measure w/ equations (example 1)

We want a minor arc but we only know the two equations there, so: Because of the placement of (4K + 159) and (2k + 153) we can infer that they are vertical angles, meaning both of those equations make the same thing. We can also write that as 4K + 159 = 2k + 153 and solve it as your average problem: 2k = -6 K = -3 Now that we know what K is, just substitute it into one of the original equations, like so: 4(-3) + 159 -12 + 159 147. The measure of our minor arc is 147 (on both sides bc vertical angles)

Finding arc length

center angle/360º= arc length/2(3.14)r Circumference = 2(pi)Radius

Note about quadrilaterals

literally any one of them has all its interior angles add up to 360

converting degrees to radians

multiply by pi/180, then simplify the resulting fraction. No mixed fractions

Circle definition

the set of all points in a plane that are equidistant from a given point

Minor arc

the shortest arc connecting two endpoints on a circle (Red)