GEOMETRY

Objectives

---

Students will:
(1.) Explain the basic terms in Geometry.
(2.) Discuss the concepts of lines, line segments, and angles.
(3.) Name, classify, and measure angles.
(4.) Discuss geometric shapes.
(5.) Determine the measures of the interior and exterior angles of polygons.
(6.) Solve problems involving lines and angles.
(7.) Solve problems involving geometrical shapes.

Symbols

---

Formulas

---

(1.) $\overleftrightarrow{AB}$
Line AB

(2.) $\overline{AB}$
Line Segment AB

(3.) $\overrightarrow{AB}$
Ray AB

(4.) Given: n collinear points:
the number of different rays that can be named is: 2(n − 1) rays.

(5.) Given: n collinear points:
the number of different lines that can be named if order is important is found from the Permutation formula.
Two points are used to name a line.
Permutation: order is important: perm 2 from n
$P(n, 2) = \dfrac{n!}{(n - 2)!}$

(6.) Given: n collinear points:
the number of different lines that can be named if order is not important is found from the Combination formula.
Two points are used to name a line.
Combination: order is not important: comb 2 from n
$C(n, 2) = \dfrac{n!}{(n - 2)! * 2!}$

(7.) Given: n coplanar points:
the number of different ways to name the plane if order is important is found from the Permutation formula.
Three points are used to name a plane.
Permutation: order is important: perm 3 from n
$P(n, 3) = \dfrac{n!}{(n - 3)!}$

(8.) Given: n coplanar points:
the number of different ways to name the plane if order is not important is found from the Combination formula.
Three points are used to name a plane.
Combination: order is not important: comb 3 from n
$C(n, 3) = \dfrac{n!}{(n - 3)! * 3!}$

(9.) 12-hour format Clock: (modulo 12 Clock):
m = number of minutes
h = number of hours

(A.) Minute hand angle = 6m
(B.) Hour hand angle = 30h + 0.5m
(C.) Angle Between Hour hand and Minute hand:

$ 1st\;\;\angle = |30h - 5.5m| \\[3ex] 2nd\;\;\angle = 360 - |30h - 5.5m| $

(10.) For a polygon of n​ sides:
the sum of the interior angles of the polygon is 180(n − 2) OR 90(2n − 4)

(11.) Each Interior Angle of a Regular Polygon = $\dfrac{180(n - 2)}{n}$

(12.) Euler's Theorem: The number of faces (F ), vertices (V ), and edges (E ) of a polyhedron are related by the formula: F + V = E + 2
⇒ F + V − E = 2

Basic Geometry

---

In naming a line, two points are used.
In naming a plane, three points are used.

We can measure angles in:
Degrees (DEG, $^\circ$)
Radians (RAD)
Gradians (GRAD)
Degrees, Minutes, and Seconds ($^\circ \:'\:''$)

$DRG$ means $Degree-Radian-Gradian$ in some calculators
$180^\circ = \pi \:\:RAD = 200 \:\:GRAD$

To convert from:
radians to degrees, multiply by $\dfrac{180}{\pi}$

degrees to radians, multiply by $\dfrac{\pi}{180}$

$DMS$ means $Degree-Minute-Second$ in some calculators

$ 1^\circ = 60' \\[3ex] 1^\circ = 3600'' \\[3ex] 1^\circ = 60' = 3600'' \\[3ex] 1' = 60'' \\[3ex] $ To convert from:
degrees to minutes, multiply by $60$
minutes to degrees, divide by $60$
degrees to seconds, multiply by $3600$
seconds to degrees, divide by $3600$
minutes to seconds, multiply by $60$
seconds to minutes, divide by $60$

Show students these angular measures in their scientific calculators.
Show them how to convert from one angular measure to another.

Notable Notes

(1.) Only 1 line can contain a particular segment because a line segment has 2 endpoints and 2 points determine a line.

(2.) A ray can contain an infinite number because there are an infinite number of points on a line and any point not on the segment that would be on the line containing the line​ segment, can be the endpoint of a ray containing the segment.

(3.) An infinite number of planes can pass through a given line.

(4.) It is not possible for three points to be noncoplanar because if the points are collinear they can be contained in an infinite number of planes and if the points are​ noncollinear, exactly one plane contains them.

(5.) Given a line and a point not on the​ line, there is only one plane containing the point and the line. This is the explanation.
A plane is uniquely determined by three noncollinear points. Any two of these points determines a unique​ line, so that any plane containing the line and the remaining point not on the line is uniquely determined.

(6.) Compare and Contrast: Parallel lines and Skew lines
Compare: Parallel lines and skew lines are similar in that they do not intersect.
Contrast: Parallel lines and skew lines are different in that parallel lines are coplanar and skew lines are not coplanar.

(7.) If planes α and β are distinct planes having points​ A, B, and C in​ common:
then points​ A, B, and C are collinear because if two distinct planes​ intersect, the intersection is a line and the common points must all lie on that line.

(8.) It is not possible to locate four points in a plane such that the number of lines determined by the points is not exactly​ 1, 4, or 6.
In other words, to locate four points in a plane, the number of lines determined by the points is exactly​ 1, 4, or 6.

(9.) Examples of physical objects that represents:
(a.) A right angle is the edge of a table and one leg of that table meeting that edge
(b.) Parallel lines are the opposite edges of the side of a table
(c.) Skew lines are a horizontal and vertical edge of a cube that do not share a vertex
(d.) Parallel planes are the covers of a book

(10.) Using a 12-hour format Clock: (modulo 12 Clock):
There are at least two hands: the hour hand (usually the short hand) and the minute hand (usually the long hand)
These two hands move at different speeds.
There are 12 numbers: clockwise-direction: 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
Between each consecutive number: 12 and 1; 1 and 2; 2 and 3; etc., there are 5 lines
Each line represents one minute
This implies that 60 lines represents 60 minutes which represents an hour
The minute hand will move a complete circle (360°) before the hour hand moves from one number to the next number.

Example: Say we move from: 12:00 pm to 1:00 pm (1 hour)
Minute hand:
moves 60 lines (60 minutes): from 12 to 12
completes 360° in 1 hour
completes 360° in 60 minutes (1 hour = 60 minutes)

$ 60\; minutes \rightarrow 360^\circ \\[3ex] 1\; minute \rightarrow \dfrac{360}{60} \\[5ex] 1\; minute \rightarrow 6^\circ \\[3ex] Speed = 6^\circ\;\;per\;\;minute \\[5ex] $ The speed of the minute hand is 6° per minute.
After 1 minute, the minute hand moves 6°
After 2 minutes, the minute hand moves 2 * 6 = 12°
After 3 minutes, the minute hand moves 3 * 6 = 18°
After m minutes, the minute hand moves m * 6 = 6m
This implies that the angle between the Minute hand and Number 12 line is: the number of minutes * 6°
Minute hand angle = 6m

Hour hand:
12-hour format: 12 numbers: 12 sections
1 hour = 1 section

$ 12\;\;sections \rightarrow 360^\circ \\[3ex] 1\;\;section \rightarrow \dfrac{360^\circ}{12} \\[5ex] 1\;\;section \rightarrow 30^\circ \\[3ex] $ ⇒
completes 30° in 1 hour
completes 360° in 12 hours

$ 1\;hour \rightarrow 30^\circ \\[3ex] 60\;minutes \rightarrow 30^\circ \\[3ex] 1 minute \rightarrow \dfrac{30}{60} \\[5ex] 1 minute \rightarrow 0.5^\circ \\[3ex] Speed = 0.5^\circ\;\;per\;\;minute \\[5ex] $ The speed of the hour hand is 0.5° per minute.
After 1 minute, the hour hand moves 0.5°
After 2 minutes, the hour hand moves 2 * 0.5 = 1°
After 3 minutes, the hour hand moves 3 * 0.5 = 1.5°
After m minutes, the hour hand moves m * 0.5 = 0.5m
This implies that the angle between the Hour hand and Number 12 line is: the number of minutes * 0.5°
Angle in Minutes = 0.5m
Also:
The speed of the hour hand is 30° per hour.
After 1 hour, the hour hand moves 30°
After 2 hours, the hour hand moves 2 * 30 = 60°
After 3 hours, the hour hand moves 3 * 30 = 90°
After h hours, the hour hand moves h * 30 = 30h
This implies that the angle between the Hour hand and Number 12 line is: the number of hours * 30°
Angle in hours = 30h

Hour hand angle = 30h + 0.5m

Angle Between Hour hand and Minute hand
There are two angles between the Hour hand and the Minute hand.
First angle = absolute value of the difference between the Hour hand angle and the Minute hand angle
Second angle = difference between 360° and the First angle.

In Summary:
m = number of minutes
h = number of hours

(A.) Minute hand angle = 6m
(B.) Hour hand angle = 30h + 0.5m
(C.) Angle Between Hour hand and Minute hand:
(i.) First angle = |Hour hand angle − Minute hand angle|

$ 1st\;\;\angle = |30h + 0.5m - 6m| \\[3ex] 1st\;\;\angle = |30h - 5.5m| \\[3ex] $ (ii.) Second angle = 360 − First angle

$ 2nd\;\;\angle = 360 - |30h - 5.5m| $

Polygons

---

A polygon is a closed plane shape with straight sides.
Poygons are classified based on their number of sides.
Because it must be a closed shape with straight sides, the smallest polygon we can have is a shape with 3 straight sides. This is a 3-gon also known as a triangle.

Classification of Polygons

Name	Other Name(s)	Number of Sides (n)	Examples	Sum of Interior Angles 180(n − 2)	Each Interior Angle of a Regular Polygon $\dfrac{180(n - 2)}{n}$
Triangle	3-gon	3	Side Classification Scalene Triangle Isosceles Triangle Equilateral Triangle Angle Classification Acute Triangle Right Triangle Obtuse Triangle Oblique Triangle	$ 180(3 - 2) \\[3ex] 180 (1) \\[3ex] 180^\circ $	Each Interior angle of a Regular Triangle (Equilateral Triangle) $ \dfrac{180}{3} \\[5ex] 60^\circ $
Quadrilateral	4-gon Tetragon	4	Square Rectangle Parallelogram Rhombus Kite Trapezoid	$ 180(4 - 2) \\[3ex] 180(2) \\[3ex] 360^\circ $	$ \dfrac{360}{4} \\[5ex] 90^\circ $
Pentagon	5-gon	5		$ 180(5 - 2) \\[3ex] 180(3) \\[3ex] 540^\circ $	$ \dfrac{540}{5} \\[5ex] 108^\circ $
Hexagon	6-gon	6		$ 180(6 - 2) \\[3ex] 180(4) \\[3ex] 720^\circ $	$ \dfrac{720}{6} \\[5ex] 120^\circ $
Heptagon	7-gon	7		$ 180(7 - 2) \\[3ex] 180(5) \\[3ex] 900^\circ $	$ \dfrac{900}{7} \\[5ex] 128.5714286^\circ $
Octagon	8-gon	8		$ 180(8 - 2) \\[3ex] 180(6) \\[3ex] 1080^\circ $	$ \dfrac{1080}{8} \\[5ex] 135^\circ $
Nonagon	9-gon	9		$ 180(9 - 2) \\[3ex] 180(7) \\[3ex] 1260^\circ $	$ \dfrac{1260}{9} \\[5ex] 140^\circ $
Decagon	10-gon	10		$ 180(10 - 2) \\[3ex] 180(8) \\[3ex] 1440^\circ $	$ \dfrac{1440}{10} \\[5ex] 144^\circ $
Hendecagon	11-gon	11		$ 180(11 - 2) \\[3ex] 180(9) \\[3ex] 1620^\circ $	$ \dfrac{1620}{11} \\[5ex] 147.2727273^\circ $
Dodecagon	12-gon	12		$ 180(12 - 2) \\[3ex] 180(10) \\[3ex] 1800^\circ $	$ \dfrac{1800}{12} \\[5ex] 150^\circ $
Tridecagon	13-gon	13		$ 180(13 - 2) \\[3ex] 180(11) \\[3ex] 1980^\circ $	$ \dfrac{1980}{13} \\[5ex] 152.3076923^\circ $
Tetradecagon	14-gon	14		$ 180(14 - 2) \\[3ex] 180(12) \\[3ex] 2160^\circ $	$ \dfrac{2160}{14} \\[5ex] 154.2857143^\circ $
Pendedecagon	15-gon	15		$ 180(15 - 2) \\[3ex] 180(13) \\[3ex] 2340^\circ $	$ \dfrac{2340}{15} \\[5ex] 156^\circ $
Hexdecagon	16-gon	16		$ 180(16 - 2) \\[3ex] 180(14) \\[3ex] 2520^\circ $	$ \dfrac{2520}{16} \\[5ex] 157.5^\circ $
Heptdecagon	17-gon	17		$ 180(17 - 2) \\[3ex] 180(15) \\[3ex] 2700^\circ $	$ \dfrac{2700}{17} \\[5ex] 158.8235294^\circ $
Octdecagon	18-gon	18		$ 180(18 - 2) \\[3ex] 180(16) \\[3ex] 2880^\circ $	$ \dfrac{2880}{18} \\[5ex] 160^\circ $
Enneadecagon	19-gon	19		$ 180(19 - 2) \\[3ex] 180(17) \\[3ex] 3060^\circ $	$ \dfrac{3060}{19} \\[5ex] 161.0526316^\circ $
Icosagon	20-gon	20		$ 180(20 - 2) \\[3ex] 180(18) \\[3ex] 3240^\circ $	$ \dfrac{3240}{20} \\[5ex] 162^\circ $

Circles

---

A Circle is defined as the locus of all points on a plane equidistant (equal distance) from a fixed point.
The fixed point is the center of the circle.
The equal distance is the radius of the circle.

• Symbols and Meanings
• (h, k) = coordinates of the center of a circle
• r = radius of a circle
• d = diameter of a circle
• C = circumference of a circle (also known as the perimeter)
• A = area of a circle
• π = pi = $\dfrac{22}{7}$
• θ = central angle
• ° = DEG = degrees
• RAD = radians
• L = length of arc
• A_sec = area of sector
• P_sec = perimeter of sector
• A_seg = area of segment
• P_seg = perimeter of segment
• (x₁, y₁) = first endpoint of the diameter of a circle
• (x₂, y₂) = second endpoint of the diameter of a circle

Formulas

(1.) Radius, Diameter, Circumference, Area

$ \underline{Circle} \\[3ex] d = 2r \\[3ex] r = \dfrac{d}{2} \\[5ex] C = \pi d \\[3ex] d = \dfrac{C}{\pi} \\[5ex] C = 2\pi r \\[3ex] r = \dfrac{C}{2\pi} \\[5ex] A = \pi r^2 \\[3ex] r = \sqrt{\dfrac{A}{\pi}} \\[5ex] A = \dfrac{\pi d^2}{4} \\[5ex] d = \sqrt{\dfrac{4A}{\pi}} \\[5ex] A = \dfrac{C^2}{4\pi} \\[5ex] C = 2\sqrt{A\pi} \\[5ex] \underline{Semicircle} \\[3ex] d = 2r \\[3ex] r = \dfrac{d}{2} \\[5ex] C = \pi r \\[3ex] C = \dfrac{\pi d}{2} \\[5ex] r = \dfrac{C}{\pi} \\[5ex] d = \dfrac{2C}{\pi} \\[5ex] A = \dfrac{\pi r^2}{2} \\[5ex] r = \sqrt{\dfrac{2A}{\pi}} \\[5ex] A = \dfrac{\pi d^2}{8} \\[5ex] d = \sqrt{\dfrac{8A}{\pi}} \\[7ex] \underline{\theta\;\;in\;\;DEG} \\[3ex] L = \dfrac{2\pi r\theta}{360} \\[5ex] \theta = \dfrac{180L}{\pi r} \\[5ex] r = \dfrac{180L}{\pi \theta} \\[5ex] A_{sec} = \dfrac{\pi r^2\theta}{360} \\[5ex] P_{sec} = \dfrac{r(\pi\theta + 360)}{180} \\[5ex] \theta = \dfrac{360A_{sec}}{\pi r^2} \\[5ex] r = \dfrac{360A_{sec}}{\pi\theta} \\[5ex] A_{sec} = \dfrac{Lr}{2} \\[5ex] A_{sec} = \dfrac{Lr}{2} \\[5ex] r = \dfrac{2A_{sec}}{L} \\[5ex] L = \dfrac{2A_{sec}}{r} \\[5ex] \underline{\theta\;\;in\;\;RAD} \\[3ex] L = r\theta \\[5ex] \theta = \dfrac{L}{r} \\[5ex] r = \dfrac{L}{\theta} \\[5ex] A_{sec} = \dfrac{r^2\theta}{2} \\[5ex] \theta = \dfrac{2A_{sec}}{r^2} \\[5ex] r = \sqrt{\dfrac{2A_{sec}}{\theta}} \\[5ex] $ (2.) Standard Form of the Equation of a Circle

$(x - h)^2 + (y - k)^2 = r^2$
where:
$x, y$ are the variables
$(h, k)$ are the coordinates of the center of the circle
$r$ is the radius of the circle

(3.) General Form of the Equation of a Circle
$x^2 + y^2 + 2gx + 2fy + c = 0$
where:
$x, y$ are the variables
$c$ is the coefficient of $x$
$d$ is the coefficient of $y$
$c, d, e$ are values/constants

(4.) Converting Between Forms

Given: Standard Form of the Equation of a Circle
To Find: General Form of the Equation of the Circle

$ \underline{General\;\;Form} \\[3ex] x^2 + y^2 + 2gx + 2fy + c = 0 \\[5ex] \underline{Standard\;\;Form} \\[3ex] (x - h)^2 + (y - k)^2 = r^2 \\[3ex] (x - h)^2 + (y - k)^2 - r^2 = 0 \\[3ex] [(x - h)(x - h)] + [(y - k)(y - k)] - r^2 = 0 \\[3ex] [x^2 - hx - hx + h^2] + [y^2 - ky - ky + k^2] - r^2 = 0 \\[3ex] (x^2 - 2hx + h^2) + (y^2 - 2ky + k^2) - r^2 = 0 \\[3ex] x^2 - 2hx + h^2 + y^2 - 2ky + k^2 - r^2 = 0 \\[3ex] x^2 + y^2 - 2hx - 2ky + h^2 + k^2 - r^2 = 0 \\[5ex] General\;\;Form = Standard\;\;Form \\[3ex] \implies \\[3ex] RHS = RHS:\;\;0 = 0 ...okay \\[3ex] LHS = LHS: \\[3ex] x^2 + y^2 + 2gx + 2fy + c = x^2 + y^2 - 2hx - 2ky + h^2 + k^2 - r^2 \\[3ex] 2gx + 2fy + c = -2hx - 2ky + h^2 + k^2 - r^2 \\[3ex] \underline{Equate\;\;the\;\;terms\;\;in\;\;x} \\[3ex] 2gx = -2hx \\[3ex] g = -h \\[3ex] \underline{Equate\;\;the\;\;terms\;\;in\;\;y} \\[3ex] 2fy = -2ky \\[3ex] f = -k \\[3ex] \underline{Equate\;\;the\;\;constants} \\[3ex] c = h^2 + k^2 - r^2 \\[5ex] $ Given: General Form of the Equation of a Circle
To Find: Standard Form of the Equation of the Circle

$ \underline{Standard\;\;Form} \\[3ex] (x - h)^2 + (y - k)^2 = r^2 \\[3ex] (x - h)^2 + (y - k)^2 - r^2 = 0 \\[3ex] [(x - h)(x - h)] + [(y - k)(y - k)] - r^2 = 0 \\[3ex] [x^2 - hx - hx + h^2] + [y^2 - ky - ky + k^2] - r^2 = 0 \\[3ex] (x^2 - 2hx + h^2) + (y^2 - 2ky + k^2) - r^2 = 0 \\[3ex] x^2 - 2hx + h^2 + y^2 - 2ky + k^2 - r^2 = 0 \\[3ex] x^2 + y^2 - 2hx - 2ky + h^2 + k^2 - r^2 = 0 \\[5ex] \underline{General\;\;Form} \\[3ex] x^2 + y^2 + 2gx + 2fy + c = 0 \\[5ex] Standard\;\;Form = General\;\;Form \\[3ex] \implies \\[3ex] RHS = RHS:\;\;0 = 0 ...okay \\[3ex] LHS = LHS: \\[3ex] x^2 + y^2 - 2hx - 2ky + h^2 + k^2 - r^2 = x^2 + y^2 + 2gx + 2fy + c \\[3ex] - 2hx - 2ky + h^2 + k^2 - r^2 = 2gx + 2fy + c \\[3ex] \underline{Equate\;\;the\;\;terms\;\;in\;\;x} \\[3ex] -2hx = 2gx \\[3ex] -h = g \\[3ex] h = -g \\[3ex] \underline{Equate\;\;the\;\;terms\;\;in\;\;y} \\[3ex] -2ky = 2fy \\[3ex] -k = f \\[3ex] k = -f \\[3ex] \underline{Equate\;\;the\;\;constants} \\[3ex] h^2 + k^2 - r^2 = c \\[3ex] h^2 + k^2 - c = r^2 \\[3ex] r^2 = h^2 + k^2 - c \\[3ex] r^2 = (-g)^2 + (-f)^2 - c \\[3ex] r^2 = g^2 + f^2 - c \\[3ex] r = \sqrt{g^2 + f^2 - c} \\[5ex] $ (5.) Given: The Center Coordinates of a Circle and an Endpoint on the Circumference of the Circle
The coordinates of the center of the circle = $(h, k)$
The endpoint on the circumference of the circle = $(x_1, y_1)$
The radius of the circle can be found by the Distance Formula
The radius of the circle = $r$
r = $\sqrt{(x_1 - h)^2 + (y_1 - k)^2}$
The diameter of the circle = $d$
The diameter of the circle is twice the radius.
$d = 2 * r$
The second endpoint of the diameter of the circle can also be found
The second endpoint of the diameter of the circle = $(x_2, y_2)$

$ x_2 = x_1 + r \\[3ex] y_2 = y_1 + r \\[3ex] (x_2, y_2) = (x_1 + r, y_1 + r) \\[3ex] $ (6.) Given: The Endpoints of the Diameter of the Circle
$(x_1, y_1)$ = first endpoint of the diameter of a circle
$(x_2, y_2)$ = second endpoint of the diameter of a circle
The center of the circle is found using the Midpoint Formula
$(h, k)$ are the coordinates of the center of the circle

$ h = \dfrac{x_1 + x_2}{2} \\[5ex] k = \dfrac{y_1 + y_2}{2} \\[5ex] $ The radius of the circle can then be found by the Distance Formula using either endpoints
Please refer to Number (3.)

(7.) Given: Any Two Points
To Find: A Point on the $y-axis$ Equidistant From the Two Points
Let the first point = $(x_1, y_1)$
and the second point = $(x_2, y_2)$
A point on the $y-axis$ equidistant from $(x_1, y_1)$ and $(x_2, y_2)$ is $(0, y)$
This implies that the distance from $(x_1, y_1)$ to $(0, y)$ should be the same distance from $(0, y)$ to $(x_2, y_2)$

Using the Distance Formula to find the distance from $(x_1, y_1)$ to $(0, y)$ gives:

$ distance1 = \sqrt{(0 - x_1)^2 + (y - y_1)^2} \\[3ex] distance1 = \sqrt{(-x_1)^2 + (y - y_1)^2} \\[3ex] distance1 = \sqrt{x_1^2 + (y - y_1)^2} \\[3ex] $ Using the Distance Formula to find the distance from $(0, y)$ to $(x_2, y_2)$ gives:

$ distance2 = \sqrt{(x_2 - 0)^2 + (y_2 - y)^2} \\[3ex] distance2 = \sqrt{(x_2)^2 + (y_2 - y)^2} \\[3ex] distance2 = \sqrt{x_2^2 + (y_2 - y)^2} \\[3ex] $ The two distances should be the same ...the word, equidistant means equal distance
So, distance 1 = distance 2

$ \sqrt{x_1^2 + (y - y_1)^2} = \sqrt{x_2^2 + (y_2 - y)^2} \\[3ex] Square\:\:both\:\:sides \\[3ex] \left(\sqrt{x_1^2 + (y - y_1)^2}\right)^2 = \left(\sqrt{x_2^2 + (y_2 - y)^2}\right)^2 \\[3ex] x_1^2 + (y - y_1)^2 = x_2^2 + (y_2 - y)^2 \\[3ex] (y - y_1)^2 - (y_2 - y)^2 = x_2^2 - x_1^2 \\[3ex] Expand \\[3ex] (y - y_1)(y - y_1) - [(y_2 - y)(y_2 - y)] = x_2^2 - x_1^2 \\[3ex] y^2 - yy_1 - yy_1 + y_1^2 - (y_2^2 - yy_2 - yy_2 + y^2) = x_2^2 - x_1^2 \\[3ex] y^2 - 2yy_1 + y_1^2 - (y_2^2 - 2yy_2 + y^2) = x_2^2 - x_1^2 \\[3ex] y^2 - 2yy_1 + y_1^2 - y_2^2 + 2yy_2 - y^2 = x_2^2 - x_1^2 \\[3ex] y^2 \:\:cancels\:\:out \\[3ex] Collect\:\:like\:\:terms\:\:in\:\:y \\[3ex] 2yy_2 - 2yy_1 = x_2^2 - x_1^2 - y_1^2 + y_2^2 \\[3ex] y(2y_2 - 2y_1) = x_2^2 + y_2^2 - x_1^2 - y_1^2 \\[3ex] y = \dfrac{x_2^2 + y_2^2 - x_1^2 - y_1^2}{2y_2 - 2y_1} \\[5ex] $ (8.) Given: Any Two Points
To Find: A Point on the $x-axis$ Equidistant From the Two Points
Let the first point = $(x_1, y_1)$
and the second point = $(x_2, y_2)$
A point on the $x-axis$ equidistant from $(x_1, y_1)$ and $(x_2, y_2)$ is $(x, 0)$
This implies that the distance from $(x_1, y_1)$ to $(x, 0)$ should be the same distance from $(x, 0)$ to $(x_2, y_2)$

Using the Distance Formula to find the distance from $(x_1, y_1)$ to $(x, 0)$ gives:

$ distance1 = \sqrt{(x - x_1)^2 + (0 - y_1)^2} \\[3ex] distance1 = \sqrt{(x - x_1)^2 + (-y_1)^2} \\[3ex] distance1 = \sqrt{(x - x_1)^2 + y_1^2} \\[3ex] $ Using the Distance Formula to find the distance from $(x, 0)$ to $(x_2, y_2)$ gives:

$ distance2 = \sqrt{(x_2 - x)^2 + (y_2 - 0)^2} \\[3ex] distance2 = \sqrt{(x_2 - x)^2 + (y_2)^2} \\[3ex] distance2 = \sqrt{(x_2 - x)^2 + y_2^2} \\[3ex] $ The two distances should be the same ...the word, equidistant means equal distance
So, distance 1 = distance 2

$ \sqrt{(x - x_1)^2 + y_1^2} = \sqrt{(x_2 - x)^2 + y_2^2} \\[3ex] Square\:\:both\:\:sides \\[3ex] \left(\sqrt{(x - x_1)^2 + y_1^2}\right)^2 = \left(\sqrt{(x_2 - x)^2 + y_2^2}\right)^2 \\[3ex] (x - x_1)^2 + y_1^2 = (x_2 - x)^2 + y_2^2 \\[3ex] (x - x_1)^2 - (x_2 - x)^2 = y_2^2 - y_1^2 \\[3ex] Expand \\[3ex] (x - x_1)(x - x_1) - [(x_2 - x)(x_2 - x)] = y_2^2 - y_1^2 \\[3ex] x^2 - xx_1 - xx_1 + x_1^2 - (x_2^2 - xx_2 - xx_2 + x^2) = y_2^2 - y_1^2 \\[3ex] x^2 - 2xx_1 + x_1^2 - (x_2^2 - 2xx_2 + x^2) = y_2^2 - y_1^2 \\[3ex] x^2 - 2xx_1 + x_1^2 - x_2^2 + 2xx_2 - x^2 = y_2^2 - y_1^2 \\[3ex] x^2 \:\:cancels\:\:out \\[3ex] Collect\:\:like\:\:terms\:\:in\:\:x \\[3ex] 2xx_2 - 2xx_1 = y_2^2 - y_1^2 - x_1^2 + x_2^2 \\[3ex] x(2x_2 - 2x_1) = x_2^2 + y_2^2 - x_1^2 - y_1^2 \\[3ex] x = \dfrac{x_2^2 + y_2^2 - x_1^2 - y_1^2}{2x_2 - 2x_1} \\[5ex] $ (9.) Given: Center and Tangent to the Line Value
To Find: Radius

$ Center = (h, k) \\[3ex] \underline{Tangent\;\;to\;\;the\;\;line:\;\;x = some\;\;value} \\[3ex] r = |h - xValue| \\[3ex] On\;\;the\;\;y-axis;\;\;all\;\;x-values\;\;are\;\;0 \\[3ex] Tangent\;\;to\;\;the\;\;y-axis:\;\; set\;\; x = 0 \\[3ex] \underline{Tangent\;\;to\;\;the\;\;line:\;\;y = some\;\;value} \\[3ex] r = |k - yValue| \\[3ex] On\;\;the\;\;x-axis;\;\;all\;\;y-values\;\;are\;\;0 \\[3ex] Tangent\;\;to\;\;the\;\;x-axis:\;\; set\;\; y = 0 \\[5ex] $ (10.) Given: Any Three Points
To Find: Center Coordinates, Radius, Equation

Please NOTE: To do this kind of question, the knowledge of 3 * 3 Linear Systems is important.

$ 1st\;\;Point = (x_1, y_1) \\[3ex] 2nd\;\;Point = (x_2, y_2) \\[3ex] 3rd\;\;Point = (x_3, y_3) \\[3ex] \underline{General\;\;Form\;\;of\;\;the\;\;Equation\;\;of\;\;a\;\;Circle} \\[3ex] x^2 + y^2 + 2gx + 2fy + c = 0 \\[3ex] $

Equation of A Circle: Notable Notes

The equation of a circle can be written in any of these two forms:

Standard Form
$(x - h)^2 + (y - k)^2 = r^2$
where:
$x, y$ are the variables
$(h, k)$ are the coordinates of the center of the circle
$r$ is the radius of the circle

General Form
$x^2 + y^2 + cx + dy + e = 0$
where:
$x, y$ are the variables
$c$ is the coefficient of $x$
$d$ is the coefficient of $y$
$c, d, e$ are values/constants

Please NOTE For:
(1.) $(x - h)^2 + (y - k)^2 = r^2$; Center = $(h, k)$, Radius = $r$
This is the Standard Form.
It is our basic reference for discussing all other forms.

(2.) $(x - h)^2 + (y + k)^2 = r^2$; Center = $(h, -k)$, Radius = $r$

(3.) $(x + h)^2 + (y - k)^2 = r^2$; Center = $(-h, k)$, Radius = $r$

(4.) $(x + h)^2 + (y + k)^2 = r^2$; Center = $(-h, -k)$, Radius = $r$

(5.) $(x - h)^2 + (y - k)^2 = e$; Center = $(h, k)$, Radius = $\sqrt{e}$

(6.) $(x - h)^2 + (y + k)^2 = e$; Center = $(h, -k)$, Radius = $\sqrt{e}$

(7.) $(x + h)^2 + (y - k)^2 = e$; Center = $(-h, k)$, Radius = $\sqrt{e}$

(8.) $(x + h)^2 + (y + k)^2 = e$; Center = $(-h, -k)$, Radius = $\sqrt{e}$

(9.) When converting from the Standard Form to the General Form, expand.
Multiply the binomials and arrange the terms in order.

(10.) When converting from the General Form to the Standard Form, the Completing the Square method is used.
Then, arrange the terms in order.

(11.) If the circle is tangent to a line, there are at least two approaches we can use to find the radius.
First Approach: Graphical Approach: The radius is the distance between the center of the circle and the tangent line
Second Approach: Algebraic Approach: The radius is the absolute value of the difference between a coordinate of the center of the circle and the respective axis of the tangent line.
It is the absolute value of a difference because it cannot have a negative value.

(12.) If the circle is tangent to the line: x = some value, then the radius is the absolute value of the difference between the x-coordinate of the center and the x-value of the line.

(13.) If the circle is tangent to the line: y = some value, then the radius is the absolute value of the difference between the y-coordinate of the center and the y-value of the line.

(14.) If the circle is tangent to the x-axis: set y = 0 because all y-values are zero on the horizontal axis (x-axis).
The radius is the absolute value of the difference between the y-coordinate of the center and zero.
This means that the radius is the absolute value of the y-coordinate of the center.

(15.) If the circle is tangent to the y-axis: set x = 0 because all x-values are zero on the vertical axis (y-axis).
The radius is the absolute value of the difference between the x-coordinate of the center and zero.
This means that the radius is the absolute value of the x-coordinate of the center.

Solid Geometry

---

Notable Notes

(1.) For a Prism whose base has n sides: (n-sided Prism)
Number of lateral faces: LF = n
Number of faces: F = n + 2
Number of vertices: V = 2n
Number of edges: E = 3n
Euler's Formula: F + V − E = 2

(2.) For a Pyramid whose base has n sides: (n-sided Pyramid)
Number of lateral faces: LF = n
Number of faces: F = n + 1
Number of vertices: V = n + 1
Number of edges: E = 2n
Euler's Formula: F + V − E = 2

(3.)

(4.)

(5.)

(6.)

(7.)

(8.)

(9.)

(10.)

Equilateral Triangle and Circle

---

• Symbols and Meanings
• $h$ = perpendicular height of the equilateral triangle
• $r$ = radius of the circle
• $b$ = base of the equilateral triangle
• $a$ = apothem of the equilateral triangle
• $A_T$ = area of the equilateral triangle
• $A_C$ = area of the circle
• $P_T$ = perimeter of the square
• $P_C$ = circumference or perimeter of the circle
• $A_r$ = the sum of the remaining areas
• $A_{ep}$ = area of each part of the remaining areas

Equilateral Triangle inscribed in a Circle
OR a
Circle circumscribed about an Equilateral Triangle

Formulas

$ h = \dfrac{3r}{2} \\[5ex] b = r\sqrt{3} \\[3ex] a = \dfrac{r}{2} \\[5ex] r = 2a \\[3ex] A_T = \dfrac{bh}{2} \\[5ex] A_C = \pi r^2 \\[3ex] \dfrac{A_T}{A_C} = \dfrac{3\sqrt{3}}{4\pi} \\[5ex] \dfrac{A_C}{A_T} = \dfrac{4\pi \sqrt{3}}{9} \\[5ex] A_r = A_C - A_T \\[3ex] A_{ep} = \dfrac{A_r}{3} $

$ h = \dfrac{b\sqrt{3}}{2} \\[5ex] b = \dfrac{2h\sqrt{3}}{3} \\[5ex] a = \dfrac{b\sqrt{3}}{6} \\[5ex] r = \dfrac{2h}{3} \\[5ex] A_T = \dfrac{3r^2\sqrt{3}}{4} \\[5ex] $

$ h = 3a \\[3ex] b = 2a\sqrt{3} \\[3ex] a = \dfrac{h}{3} \\[5ex] r = \dfrac{b\sqrt{3}}{3} \\[5ex] $

Circle inscribed in an Equilateral Triangle
OR an
Equilateral Triangle circumscribed about a Circle

Formulas

$ h = 3r \\[3ex] b = 2r\sqrt{3} \\[3ex] a = r \\[3ex] r = a \\[3ex] A_T = \dfrac{bh}{2} \\[5ex] A_C = \pi r^2 \\[3ex] \dfrac{A_T}{A_C} = \dfrac{3\sqrt{3}}{\pi} \\[5ex] \dfrac{A_C}{A_T} = \dfrac{\pi \sqrt{3}}{9} \\[5ex] A_r = A_T - A_C \\[3ex] A_{ep} = \dfrac{A_r}{3} $

$ h = \dfrac{b\sqrt{3}}{2} \\[5ex] b = \dfrac{2h\sqrt{3}}{3} \\[5ex] a = \dfrac{b\sqrt{3}}{6} \\[5ex] r = \dfrac{h}{3} \\[5ex] A_T = 3r^2\sqrt{3} \\[3ex] $

$ h = 3a \\[3ex] b = 2a\sqrt{3} \\[3ex] a = \dfrac{h}{3} \\[5ex] r = \dfrac{b\sqrt{3}}{6} \\[5ex] $

Square and Circle

---

• Symbols and Meanings
• $d$ = diameter of the circle
• $d_S$ = diagonal of the square
• $l$ = length of a side of the square (side length of the square)
• $r$ = radius of the circle
• $a$ = apothem of the square
• $A_S$ = area of the square
• $A_C$ = area of the circle
• $P_S$ = perimeter of the square
• $P_C$ = circumference or perimeter of the circle
• $A_r$ = the sum of the remaining areas
• $A_{ep}$ = area of each part of the remaining areas

Square inscribed in a Circle
OR a
Circle circumscribed about an Square

Formulas

$ d = d_S \\[3ex] l = r\sqrt{2} \\[3ex] a = \dfrac{l}{2} \\[5ex] r = \dfrac{l\sqrt{2}}{2} \\[5ex] A_S = l^2 \\[3ex] A_C = \pi r^2 \\[3ex] \dfrac{A_S}{A_C} = \dfrac{2}{\pi} \\[5ex] \dfrac{A_C}{A_S} = \dfrac{\pi}{2} \\[5ex] A_r = A_C - A_S \\[3ex] A_{ep} = \dfrac{A_r}{4} $

$ l = 2a \\[3ex] a = \dfrac{r\sqrt{2}}{2} \\[5ex] r = a\sqrt{2} \\[3ex] A_S = 2r^2 \\[3ex] $

Circle inscribed in an Square
OR an
Square circumscribed about a Circle