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Slopes and Equations of Lines

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Chap8

Slopes and Equations of Lines

Chin-Sung Lin

- Distance Formula
- Midpoint Formula
- Slope Formula
- Parallel Lines
- Perpendicular Lines

Basic Geometry Formulas

Mr. Chin-Sung Lin

Distance Formula

Mr. Chin-Sung Lin

A (x1, y1)

B (x2, y2)

Distance between two pointsA (x1, y1) andB (x2, y2) is given by distance formula

d(A, B) =√(x2 − x1)2+ (y2 − y1)2

Distance Formula

Mr. Chin-Sung Lin

Distance Formula - Example

Calculate the distance betweenA (4, 5) andB (1, 1)

Mr. Chin-Sung Lin

Distance Formula - Example

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

Mr. Chin-Sung Lin

Distance Formula - Example

Calculate the distance betweenA (9, 5) andB (1, 5)

Mr. Chin-Sung Lin

Midpoint Formula

Mr. Chin-Sung Lin

A (x1, y1)

M (x, y)

B (x2, y2)

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

22

Midpoint Formula

M = ( )

Mr. Chin-Sung Lin

Midpoint Formula - Example

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

Mr. Chin-Sung Lin

Midpoint Formula - Example

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

Mr. Chin-Sung Lin

Slope Formula

Mr. Chin-Sung Lin

A (x1, y1)

B (x2, y2)

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

Slope Formula

m=

Mr. Chin-Sung Lin

Slope Formula - Example

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

Mr. Chin-Sung Lin

Slope Formula - Example

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

5 - 1

4 - 2

= 2

m=

Mr. Chin-Sung Lin

Slope of Lines in the Coordinate Planes

Positive slope

Mr. Chin-Sung Lin

Slope of Lines in the Coordinate Planes

Negative slope

Mr. Chin-Sung Lin

Slope of Lines in the Coordinate Planes

Zero slope

Mr. Chin-Sung Lin

Slope of Lines in the Coordinate Planes

Undefined slope

Mr. Chin-Sung Lin

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

if and only if m = n

Slope and Parallel Lines

m

n

Mr. Chin-Sung Lin

Slope and Parallel Lines - Example

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

Mr. Chin-Sung Lin

Slope and Parallel Lines - Example

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

9 - 3

4 - 2

= 3

m = n =

Mr. Chin-Sung Lin

The straight lines with slopes (m) and (n) are mutually perpendicular

if and only if m · n= -1

Slope and Perpendicular Lines

n

m

Mr. Chin-Sung Lin

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

Slope and Perpendicular Lines - Example

Mr. Chin-Sung Lin

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

6 - 2

3 - 1

= 2

since m · n= -1,

2 · n = -1, so,n = -1/2

Slope and Perpendicular Lines - Example

m =

Mr. Chin-Sung Lin

Group Work

Mr. Chin-Sung Lin

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

Parallel and Perpendicular Lines

Mr. Chin-Sung Lin

Equations of Lines

Mr. Chin-Sung Lin

Linear equation can be written in slope-intercept form:

y = mx + b

where m is the slope

bis the y-intercept

Slope Intercept Form

b

slope: m

Mr. Chin-Sung Lin

Given: If the slope of a line is 3 and it passes through(0, 2), write the equation of the line in slope-intercept form

Write Slope Intercept Form

Mr. Chin-Sung Lin

Given: If the slope of a line is 3 and it passes through(0, 2), write the equation of the line in slope-intercept form

m = 3, b = 2

y = 3x + 2

Write Slope Intercept Form

Mr. Chin-Sung Lin

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

Write Slope Intercept Form

(0, b)

(x1, y1)

Mr. Chin-Sung Lin

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 line

Step 2: Find the y-intercept b

Step 3: Write the equation

y= mx + b

Write Slope Intercept Form

(0, b)

(x1, y1)

Mr. Chin-Sung Lin

Given: Two points (0, 4) and (2, 0)

Write Slope Intercept Form

(0, 4)

(2, 0)

Mr. Chin-Sung Lin

Given: 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) = -2

Step 2: Find the y-intercept: b = 4

Step 3: Write the equation:

y = -2x + 4

Write Slope Intercept Form

(0, 4)

(2, 0)

Mr. Chin-Sung Lin

Write Slope Intercept Form - Example

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

Mr. Chin-Sung Lin

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

Point-Slope Form

(x1, y1)

slope: m

Mr. Chin-Sung Lin

Given: If the slope of a line is 3 and it passes through(5, 2), write the equation of the line in slope-intercept form

Write Point-Slope Form

Mr. Chin-Sung Lin

Given: If the slope of a line is 3 and it passes through(5, 2), write the equation of the line in slope-intercept form

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

y - 2 = 3(x – 5)

Write Point-Slope Form

Mr. Chin-Sung Lin

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

Write Point-Slope Form

(x1, y1)

(x2, y2)

Mr. Chin-Sung Lin

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)

Write Point-Slope Form

(x1, y1)

(x2, y2)

Mr. Chin-Sung Lin

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

Write Point-Slope Form Example

(1, 4)

(3, 1)

Mr. Chin-Sung Lin

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)

= -3/2

Step 2: Write the equation

y – 1 = (-3/2)(x – 3)

Write Point-Slope Form Example

(1, 4)

(3, 1)

Mr. Chin-Sung Lin

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

Write Point-Slope Form Example

Mr. Chin-Sung Lin

Equations of Parallel & Perpendicular Lines

Mr. Chin-Sung Lin

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

Equation of a Parallel Line

Mr. Chin-Sung Lin

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 = 3

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

Equation of a Parallel Line

Mr. Chin-Sung Lin

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

Equation of a Parallel Line - Example

Mr. Chin-Sung Lin

Write an equation of the line passing through the point (2, 0) that is parallel to the line y = x - 2

Equation of a Parallel Line

Mr. Chin-Sung Lin

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

Equation of a Perpendicular Line

Mr. Chin-Sung Lin

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 = 2

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

Equation of a Perpendicular Line

Mr. Chin-Sung Lin

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

Equation of a Perpendicular Line

Mr. Chin-Sung Lin

Write an equation of the line passing through the point (4, 1)that is perpendicular to the line y = -x+ 2

Equation of a Perpendicular Line

Mr. Chin-Sung Lin

Group Work

Mr. Chin-Sung Lin

Based on the information in the graph, write the equations of line P and line Q in both slope-intercept form and point-slope form

Equation of a Perpendicular Line

Q

K

y = 2x -5

P

(4, 3)

-2

Mr. Chin-Sung Lin

Coordinate Proof

Mr. Chin-Sung Lin

- Two types of proofs in coordinate geometry:
- Special cases
- Given ordered pairs of numbers, and prove something about a specific segment or polygon
- General Theorems
- When 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

Coordinate Proof

Mr. Chin-Sung Lin

- Two skills of proofs in coordinate geometry:
- Line segments bisect each other
- the midpoints of each segment are the same point
- Two lines are perpendicular to each other
- the slope of one line is the negative reciprocal of the slope of the other

Coordinate Proof

Mr. Chin-Sung Lin

If 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

Coordinate Proof – Special Cases

Mr. Chin-Sung Lin

The 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.

Coordinate Proof – Special Cases

Mr. Chin-Sung Lin

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

Coordinate Proof – Special Cases

Mr. Chin-Sung Lin

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

Aim: Coordinate Proof DoNow:

Mr. Chin-Sung Lin

If 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?

Coordinate Proof – Special Cases

Mr. Chin-Sung Lin

- Vertices definition in coordinate geometry:
- Any triangle— (a, 0), (0, b), (c, 0)

Coordinate Proof – General Theorems

(0, b)

(0, b)

(c, 0)

(a, 0)

(c, 0)

(a, 0)

Mr. Chin-Sung Lin

- Vertices definition in coordinate geometry:
- Right triangle— (a, 0), (0, b), (0, 0)

Coordinate Proof – General Theorems

(0, b)

(0, 0)

(a, 0)

Mr. Chin-Sung Lin

- Vertices definition in coordinate geometry:
- Isosceles triangle— (-a, 0), (0, b), (a, 0)

Coordinate Proof – General Theorems

(0, b)

(a, 0)

(-a, 0)

Mr. Chin-Sung Lin

- Vertices definition in coordinate geometry:
- Midpoint of segments— (2a, 0), (0, 2b), (2c, 0)

Coordinate Proof – General Theorems

(0, 2b)

(2c, 0)

(2a, 0)

Mr. Chin-Sung Lin

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

Coordinate Proof – General Theorems

Mr. Chin-Sung Lin

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

Coordinate Proof – General Theorems

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

B (0, 2b)

M

A (2a, 0)

C(0, 0)

Mr. Chin-Sung Lin

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

Coordinate Proof – General Theorems

B (0, 2b)

M

A (2a, 0)

C(0, 0)

Mr. Chin-Sung Lin

Concurrence of the Altitudes of a Triangle

Mr. Chin-Sung Lin

Orthocenter: The altitudes of a triangle intersect in one point

Acute Triangle

Altitude Concurrence - Orthocenter

B(0, b)

C(c, 0)

A(a, 0)

Mr. Chin-Sung Lin

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

Acute Triangle

Altitude Concurrence - Orthocenter

B(0, b)

C(c, 0)

A(a, 0)

Mr. Chin-Sung Lin

Orthocenter: The altitudes of a triangle intersect in one point

Right Triangle

Altitude Concurrence - Orthocenter

B(0, b)

C(c, 0)

A(a, 0)

Mr. Chin-Sung Lin

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

Right Triangle

Altitude Concurrence - Orthocenter

B(0, b)

C(c, 0)

A(a, 0)

Mr. Chin-Sung Lin

Orthocenter: The altitudes of a triangle intersect in one point

Obtuse Triangle

Altitude Concurrence - Orthocenter

B(0, b)

C(c, 0)

A(a, 0)

Mr. Chin-Sung Lin

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

Obtuse Triangle

Altitude Concurrence - Orthocenter

B(0, b)

C(c, 0)

A(a, 0)

Mr. Chin-Sung Lin

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

Orthocenter

Mr. Chin-Sung Lin

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)

Orthocenter

Mr. Chin-Sung Lin

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

Orthocenter

Mr. Chin-Sung Lin

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)

Orthocenter

Mr. Chin-Sung Lin

The End

Mr. Chin-Sung Lin