slopes and equations of lines

Slopes and Equations of Lines

Chin-Sung Lin

Chap 8

Basic Geometry Formulas

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Distance Formula

Midpoint Formula

Slope Formula

Parallel Lines

Perpendicular Lines

Distance Formula

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Distance Formula

Distance between two points A (x1, y1) and B (x2, y2) is given by distance formula

d(A, B) =√(x2 − x1 )2 + (y2 − y1 )2

A (x1, y1) B (x2, y2)

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Distance Formula - Example

Calculate the distance between A (4, 5) and B (1, 1)

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Calculate the length of AB if the coordinates of A and B are (4, 15) and (-1, 3) respectively

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Calculate the distance between A (9, 5) and B (1, 5)

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Midpoint Formula

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Midpoint Formula

If the coordinates of A and B are ( x1, y1) and ( x2, y2) respectively, then the midpoint, M, of AB is given by the midpoint formula

x1 + x2, y1+ y2

2 2M = ( )

A (x1, y1) B (x2, y2)M (x, y)

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Midpoint Formula - Example

Calculate the midpoint of AB if the coordinates of A and B are (2, 7) and (-6, 5) respectively

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Midpoint Formula - Example

M(1, -2) is the midpoint of AB and the coordinates of A are (-3, 2). Find the coordinates of B

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Slope Formula

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Slope Formula

If the coordinates of A and B are (x1, y1) and (x2, y2) respectively, then the slope, m, of AB is given by the slope formula

y2 - y1

x2 - x1

m =

A (x1, y1

)

B (x2, y2

)

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Slope Formula - Example

Calculate the slope of AB, where A (4, 5) and B (2, 1)

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Slope Formula - Example

Calculate the slope of AB, where A (4, 5) and B (2, 1)

5 - 1

4 - 2

= 2

m =

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Slope of Lines in the Coordinate Planes

Positive slope

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Negative slope

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Zero slope


Undefined slope

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Slope and Parallel Lines

The straight lines with slopes (m) and (n) are parallel to each other if and only if m = n

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m n

Slope and Parallel Lines - Example

If AB is parallel to CD where A (2, 3) and B (4, 9), calculate the slope of CD

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Slope and Parallel Lines - Example

If AB is parallel to CD where A (2, 3) and B (4, 9), calculate the slope of CD

9 - 3

4 - 2

= 3

m = n =

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Slope and Perpendicular Lines

The straight lines with slopes (m) and (n) are mutually perpendicular if and only if m · n = -1

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mn

Slope and Perpendicular Lines - Example

If AB is perpendicular to CD where A (1, 2) and B (3, 6), calculate the slope of CD

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Slope and Perpendicular Lines - Example

If AB is perpendicular to CD where A (1, 2) and B (3, 6), calculate the slope of CD

6 - 2

3 - 1

= 2

since m · n = -1, 2 · n = -1, so, n = -1/2

m =

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Group Work

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Parallel and Perpendicular Lines

There are four points A (2, 6), B(6, 4), C(4, 0) and D(0, 2) on the coordinate plane.

Identify the pairs of parallel and perpendicular lines

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Equations of Lines

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Slope Intercept Form

Linear equation can be written in slope-intercept form:

y = mx + bwhere m is the slope

b is the y-intercept

slope: m

b

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Write Slope Intercept FormGiven: If the slope of a line is 3 and it passes

through(0, 2), write the equation of the line in slope-intercept form

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Write Slope Intercept FormGiven: If the slope of a line is 3 and it passes


m = 3, b = 2y = 3x + 2

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Write Slope Intercept Form

Given: y-intercept b and a point (x1, y1)

(0, b)

(x1, y1)

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Write Slope Intercept Form

Given: y-intercept b and a point (x1, y1)

Step 1: Find the slope m by choosing two points (0, b) and (x1, y1) on the graph of the

lineStep 2: Find the y-intercept bStep 3: Write the equation

y = mx + b

(0, b)

(x1, y1)

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Write Slope Intercept FormGiven: Two points (0, 4) and (2, 0)

(0, 4)

(2, 0)

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Write Slope Intercept FormGiven: Two points (0, 4) and (2, 0)

Step 1: Find the slope by choosing two points on the graph of the line: m = (0-4)/(2-0)

= -2Step 2: Find the y-intercept: b = 4Step 3: Write the equation:

y = -2x + 4

(0, 4)

(2, 0)

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Write Slope Intercept Form - Example

A line passing through (2, 3) and the y-intercept is -5. Write the equation

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Point-Slope Form

Linear equation can be written in point-slope form:

y – y1 = m(x – x1)where m is the slope

(x1, y1) is a point on the line

slope: m

(x1, y1)

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Write Point-Slope FormGiven: If the slope of a line is 3 and it passes


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Write Point-Slope FormGiven: If the slope of a line is 3 and it passes


m = 3, (x1, y1) = (5, 2)

y - 2 = 3(x – 5)

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Write Point-Slope Form

Given: Two points (x1, y1) and (x2, y2)

(x1, y1)

(x2, y2)

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Write Point-Slope Form

Given: Two points (x1, y1) and (x2, y2)

Step 1: Find the slope m by plugging two points (x1, y1) and (x2, y2) into the slop formula m = (y2 – y1)/(x2 – x1)

Step 2: Write the equation using slope m and any point y – y1 = m(x – x1)

(x1, y1)

(x2, y2)

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Write Point-Slope Form Example

Given: Two points (3, 1) and (1, 4)

(1, 4)

(3, 1)

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Given: Two points (3, 1) and (1, 4)Step 1: Find the slope m by plugging two

points (3, 1) and (1, 4) into the slop formula m = (4 – 1)/(1 – 3)

= -3/2Step 2: Write the equation

y – 1 = (-3/2)(x – 3)

(1, 4)

(3, 1)

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Given: Two points (-2, 7) and (2, 3)

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Equations of Parallel & Perpendicular Lines

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Equation of a Parallel Line

Write an equation of the line passing through the point (-1, 1) that is parallel to the line y = 2x – 3

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Equation of a Parallel Line

Write an equation of the line passing through the point (-1, 1) that is parallel to the line y = 2x - 3

Step 1: Find the slope m from the given equation: since two lines are parallel, the slopes are

the same, so: m = 2

Step 2: Find the y-intercept b by using the m = 2 and the given point (-1, 1): 1 = 2(-1) + b, so, b

= 3Step 3: Write the equation: y = 2x + 3

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Equation of a Parallel Line - Example

Write an equation of the line passing through the point (2, 3) that is parallel to the line y = x – 5

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Equation of a Parallel LineWrite an equation of the line passing through the

point (2, 0) that is parallel to the line y = x - 2

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Equation of a Perpendicular Line

Write an equation of the line passing through the point (2, 3) that is perpendicular to the line y = -2x + 2

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Equation of a Perpendicular Line

Write an equation of the line passing through the point (2, 3) that is perpendicular to the line y = -2x + 2

Step 1: Find the slope m from the given equation: since two lines are perpendicular, the

product of the slopes is equal to -1, so: m = 1/2

Step 2: Find the y-intercept b by using the m = 2 and the given point (-1, 1): 3 = (1/2)(2) + b, so,

b = 2Step 3: Write the equation: y = (1/2)x + 2

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Equation of a Perpendicular LineWrite an equation of the line passing through the

point (1, 2) that is perpendicular to the line y = x + 3

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Equation of a Perpendicular LineWrite an equation of the line passing through the

point (4, 1) that is perpendicular to the line y = -x + 2

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Group Work

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Equation of a Perpendicular LineBased on the information in the graph, write the

equations of line P and line Q in both slope-intercept form and point-slope form

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(4, 3)

y = 2x -5

-2

K

P

Q

Coordinate Proof

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Coordinate Proof

Two types of proofs in coordinate geometry:• Special cases

Given ordered pairs of numbers, and prove something about a specific segment or polygon

• General TheoremsWhen the given information is a figure that represents a particular type of polygon, we must state the coordinates of its vertices in general terms using variables

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Coordinate Proof

Two skills of proofs in coordinate geometry:

• Line segments bisect each otherthe midpoints of each segment are the same point

• Two lines are perpendicular to each otherthe slope of one line is the negative reciprocal of the slope of the other

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Coordinate Proof – Special CasesIf the coordinates of four points are A(-3, 5), B(5,

1), C(-2, -3), and D(4, 9), prove that AB and CD are perpendicular bisector to each other

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Coordinate Proof – Special CasesThe vertices of rhombus ABCD are A(2, -3), B(5, 1),

C(10, 1) and D(7, -3). (a) Prove that the diagonals bisect each other. (b) Prove that the diagonals are perpendicular to each other.

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Coordinate Proof – Special CasesIf the coordinates of three points are A(-1, 4), B(4,

7), and C(1, 2), prove that ABC is an isosceles triangle

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Aim: Coordinate Proof DoNow:

If the coordinates of three points are A(-1, 4), B(4, 7), and C(1, 2), prove that ABC is an isosceles triangle

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Coordinate Proof – Special CasesIf the coordinates of three points are A(4, 3), B(6, 7),

and C(-4, 7), prove that ΔABC is a right triangle. Which angle is the right angle?

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Coordinate Proof – General Theorems

Vertices definition in coordinate geometry:

• Any triangle— (a, 0), (0, b), (c, 0)

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(0, b)

(a, 0) (c, 0)

(0, b)

(a, 0) (c, 0)



• Right triangle— (a, 0), (0, b), (0, 0)

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(0, b)

(0, 0) (a, 0)



• Isosceles triangle— (-a, 0), (0, b), (a, 0)

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(0, b)

(-a, 0) (a, 0)



• Midpoint of segments— (2a, 0), (0, 2b), (2c, 0)

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(0, 2b)

(2a, 0) (2c, 0)


Prove that the midpoint of the hypotenuse of a right triangle is equidistance from the vertices

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B (0, 2b)

A (2a, 0)

C(0, 0)

M

Given: Right triangle ABC whose vertices are A(2a, 0), B(0, 2b), and C(0,0). Let M be the midpoint of the hypotenuse AB

Prove: AM = BM = CM



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B (0, 2b)

A (2a, 0)C(0, 0)

M

Concurrence of the Altitudes of a Triangle

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Altitude Concurrence - Orthocenter

Orthocenter: The altitudes of a triangle intersect in one point

Acute Triangle

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B(0, b)

A(a, 0) C(c, 0)


Theorem: The altitudes of a triangle are concurrent (intersecting in one point— orthocenter)

Acute Triangle

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B(0, b)

A(a, 0) C(c, 0)



Right Triangle

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B(0, b)

A(a, 0) C(c, 0)



Obtuse Triangle

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B(0, b)

A(a, 0) C(c, 0)

Orthocenter

The coordinates of the vertices of ΔABC are A(0, 0), B(-2, 6), and C(4, 0). Find the coordinates of the orthocenter of the triangle

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Orthocenter

The coordinates of the vertices of ΔABC are A(0, 0), B(-2, 6), and C(4, 0). Find the coordinates of the orthocenter of the triangle

Answer: (-2, -2)

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Orthocenter

The coordinates of the vertices of ΔABC are A(0, 0), B(3, 4), and C(2, 1). Find the coordinates of the orthocenter of the triangle

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Orthocenter

The coordinates of the vertices of ΔABC are A(0, 0), B(3, 4), and C(2, 1). Find the coordinates of the orthocenter of the triangle

Answer: (6, -2)

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The End

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slopes and equations of lines

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