Homework Section 13-1 - St. Charles Preparatory School Links... · Coordinate Geometry USE GRAPH PAPER FOR HOMEWORK IN THIS CHAPTER, ... lines, and circles on a ... 3 (-1, -4) (3,

Chapter Thirteen

Coordinate Geometry

USE GRAPH PAPER FOR HOMEWORK IN THIS CHAPTER, AS NECESSARY!

Make sure you properly annotate all coordinate graphs in this chapter!

Objectives

A. Use the terms defined in the chapter

correctly.

B. Properly use and interpret the symbols

for the terms and concepts in this chapter.

C. Appropriately apply the postulates and

theorems in this chapter.

D. Understand and apply the distance and

midpoint formulas.

E. Calculate and use the slopes of lines.

F. Perform the basics of vector mathematics.

G. Graph linear equations

H. Write the equations of straight line graphs

I. Organize and write a coordinate proof.

Section 13-1

The Distance Formula

Homework Pages 526-527:

1-24, 28, 36

Exclude 14, 16

Objectives

A. Understand and apply the terms

‘origin’, ‘axes’, ‘quadrants’, and

‘coordinate plane’.

B. Properly graph points, lines, and

circles on a coordinate plane.

C. Derive and utilize the distance

formula.

D. Understand the components of the equation of a circle.

E. Graph a circle in a coordinate plane based on its equation.

• coordinate plane: plane formed by the intersection of a

horizontal and a vertical real number line, called the

coordinate axes, where every point in the plane can be

represented by an ordered pair of real numbers, called its

coordinates.

• x-axis: the horizontal coordinate axis

• y-axis: the vertical coordinate axis

• origin: intersection of the coordinate axes

The Coordinate Plane

Coordinate Plane

1

3

1

2

2 3 -1

-2

-2

x-axis

y-axis

origin

8

5 (8, 5)

Quadrant I:

Both x & y are positive

Quadrant II:

x is negative; y is positive

Quadrant III:

Both x & y are negative

Quadrant IV:

x is positive; y is negative

y

x

The Coordinate Plane - Quadrants

Quadrants: four regions of the coordinate plane created by the two axes.

Building the Distance Formula

1 2 3 -1 -2

-2

1

3 2

-2

6

1. Label the points.

P1

P2

2. Label the coordinates

(x1,y1) (x2,y2)

P2=(x2,y2)

P1=(x1,y1)

P2=(6,3)

P1=(3,2)

3. Substitute values

Distance between points

on x-axis, |x2 – x1|

|y2 – y1|

Building the Distance Formula

(3,2)

(6,3)

|x2 – x1|

|y2 – y1|

22

222

bac

bac

212

2

12 yyxxc

Theorem 13-1 (Distance Formula)

The distance d between points (x1, y1 ) and (x2, y2 ) is given

by: d x x y y 2 12

2 12

(x2 , y2)

(x1 , y1)

d y2 - y1

x2 - x1

2122

122 yyxxd

Theorem 13-2 (Standard Form for the equation of a Circle)

An equation of a circle with center (a, b) and radius r

is (x - a)2 + (y - b)2 = r2.

(x, y)

(a, b)

r

x - a

y - b

Sample Problems

Find the distance between the points. If necessary, you may draw graphs but you should NOT need to use the distance formula.

1. (- 2, - 3) & (- 2, 4) Plot the points. What should you do first?

1 2 3

y

x

1

2

3

(-2, -3)

(-2, 4)

What is the distance between these 2 points?

7 units

Sample Problems

Find the distance between the points. If necessary, you may draw graphs but you should NOT need to use the distance formula.

3. (3, - 4) & (- 1, - 4) Plot the points. What should you do first?

What is the distance between these 2 points?

1 2 3

y

x

1

2

3

(3, -4) (-1, -4)

4 units

Sample Problems

Use the distance formula to find the distance between the 2 points.

5. (- 6, - 2) & (- 7, - 5) What should you do first?

1 2 3

y

x

1

2

3

(-7, -5)

(-6, -2)

P1 = ? P2 = ?

P1 = (- 6, - 2) P2 = (-7 , - 5)

x1 = ? y1 = ? x2 = ? y2 = ?

x1 = - 6, y1 = - 2, x2 = - 7, y2 = - 5

What is the distance formula?

d x x y y 2 12

2 12

2 2

7 6 5 2d

2 2

1 3d 1 9 1 0d u n i ts

Sample Problems

Use the distance formula to find the distance between the 2 points.

7. (- 8, 6) & (0, 0) What should you do first?

1 2 3

y

x

1

2

3

(0, 0)

(-8, 6)

P1 = ? P2 = ?

P1 = (0, 0) P2 = (- 8 , 6)

x1 = ? y1 = ? x2 = ? y2 = ?

x1 = 0, y1 = 0, x2 = - 8, y2 = 6

What is the distance formula?

d x x y y 2 12

2 12

2 2

8 0 6 0d

6 4 3 6d

1 0 0 1 0d u n i ts

A triangle with sides 6, 8, and 10. Sound familiar?

6

8

10

Sample Problems

Find the distance between the points named. Use any method

you choose.

9. (5, 4) & (1, - 2)

11. (- 2, 3) & (3, - 2)

Sample Problems

Given the points A, B & C. Find AB, BC & AC. Are A, B &

C collinear? If so, which point is in the middle?

13. A(0, 3) B(- 2, 1) C(3, 6)

15. A(- 5, 6) B(0, 2) C(3, 0)

Sample Problems

Find the center and the radius of each circle.

17. (x + 3)2 + y2 = 49

What is the Standard Form for the equation of a circle?

An equation of a circle with center (a, b) and radius r is

(x - a)2 + (y - b)2 = r2.

(x + 3)2 + y2 = 49

(x + 3)2 + (y – 0)2 = 49

x – a = x + 3 a = - 3

y – b = y - 0 b = 0 center (a, b) = center (-3, 0)

r2 = 49 radius = 7 units

Sample Problems

Find the center and the radius of each circle.

19. (x - j)2 + (y + 14)2 = 17



(x - a)2 + (y - b)2 = r2.

(x - j)2 + (y + 14)2 = 17

x – a = x - j a = j

y – b = y + 14 b = -14 center (a, b) = center (j, -14)

r2 = 17 radius = 17 units

Sample Problems

Write an equation of the circle with the given center and radius.

21. C(3, 0) r = 8 23. C(- 4, - 7) r = 5



(x - a)2 + (y - b)2 = r2.

center (3, 0) = center (a, b) a = ? b = ? a = 3, b = 0

(x - 3)2 + (y - 0)2 = 82

(x - 3)2 + y2 = 64

center (-4, -7) = center (a, b)

a = ? b = ? a = -4, b = -7

(x – (-4))2 + (y – (-7))2 = 52

(x + 4)2 + (y + 7)2 = 25

Section 13-2

Slope of a Line


1-24

Excluding 10

Objectives

A. Understand the terms ‘linear

equation’ and ‘slope of a line’.

B. Understand and identify lines with

positive, negative, zero, and

undefined slopes.

C. Calculate the slopes of various

lines.

D. Use the slope of a line to graph linear equations.

Linear Equation

• A linear equation is any equation where the graph of the solution set is a line.

• Example: y = 2x (How many solutions are there to this equation?)

SOME solutions!

If x = Then y =

(2x)

0 0

1 2

2 4

x

y

1 2 3

1

2

Plot the points. Draw the line.

How many points

are on a line?

Would it be possible to list ALL of the solutions?

Linear Equation

• NOTE! All values of x and y that satisfy the equation y = 2x form a point (x, y) that is ON the line.

• NOTE! All coordinates (x, y) of points on the line make the equation y = 2x true!

• Therefore, the GRAPH represents ALL of the solutions to the equation!

SOME solutions!

If x = Then y =

(2x)

0 0

1 2

2 4 x

y

1 2 3

1

2

Plot the points.

lines with positive slope: rise to the right

lines with negative slope: rise to the left

• steeper line: has a slope with a greater absolute value.

the slope of a horizontal line: is zero

the slope of a vertical line: is undefined

12

12

xx

yy

x

y

run

rise m slope

Slopes of Lines

rise = y = y2 - y1

run = x = x2 - x1 7

14

8 13

(13, 14)

(8, 7)

y = 14 - 7

x = 13 - 8

(x1 , y1)

(x2 , y2)

5

7

813

714

run

rise m slope

12

12

xx

yy

x

y

slope = 0

slope is undefined

12

12

xx

yy

x

y

run

rise m slope

Sample Problems

Find the slope of the line through the given points.

7. (7, 2) & (2, 7)

Label the points: P1 = (7, 2) P2 = (2, 7)

Label the coordinates of the points:

x1 = 7, y1 = 2, x2 = 2, y2 = 7

2 1

2 1

riseslope m

run

y yy

x x x

Write the formula!

2 1

2 1

7 2 5 1slope 1

2 7 5 1

y y

x x

Fill in the blanks!

1 2 3

y

x

1

2

3

P2 (2, 7)

P1 (7, 2)

What should you do first?

Sample Problems

Find the SLOPE and the length of AB

15. A(0, - 9) & B(8, - 3)

Label the points: P1 = A = (0, -9) P2 = B = (8, -3)


x1 = 0, y1 = -9, x2 = 8, y2 = -3

2 1

2 1

riseslope m

run

y yy

x x x

Write the formula!

2 1

2 1

3 9 6 3slope

8 0 8 4

y y

x x

Fill in the blanks!


y

1 2 3 x -1

-2

-3 P2 (8, -3)

P1 (0, -9)

Sample Problems

Find the slope and the LENGTH of AB

15. A(0, - 9) & B(8, - 3)

2 2

2 1 2 1distance d x x y y

Write the formula!

Fill in the blanks!

2 2

2 2

distance d 8 0 3 9

8 6 64 36 100 10

y

1 2 3 x -1

-2

-3 P2 (8, -3)

P1 (0, -9)

Label the points: P1 = A = (0, -9) P2 = B = (8, -3)


x1 = 0, y1 = -9, x2 = 8, y2 = -3

Sample Problems

Find one point to the left and to the right of the given point on

the same line.

17. P(- 3, 0) slope =

5

2

1 2 3

y

x

1

2

3

Plot the point.

If numerator of slope is

positive, go up y-units.

(if numerator negative,

go down y-units)

‘up’ 2 If denominator of slope is

positive, go right x-units.

(if denominator negative,

go left x-units)

‘right’ 5

New point (2, 2)

Sample Problems


the same line.

17. P(- 3, 0) slope =

5

2

1 2 3

y

x

1

2

3

‘up’ 2 ‘right’ 5

??5

2

5

2

If numerator of slope is

negative, go down y-units.

If denominator of slope is

negative, go left x-units.

New point (-8, -2)

Sample Problems

Find the slope of the line through the given points.

3. (1, 2) & (3, 4)

5. (1, 2) & (- 2, 5)

9. (6, - 6) & (- 6, - 6)

11. (- 4, - 3) & (- 6, - 6)

Find the slope and the length of AB

13. A(- 3, - 2) & B(7, - 6)

Sample Problems


the same line.

17. P(- 3, 0) slope =

19. P(0, - 5) slope =

Show point P, Q and R are collinear by showing PQ and QR

have the same slope.

21. P(- 8, 6) Q(- 5, 5) R(4, 2)

Complete.

23. A line with slope passes through points (2, 3) & (10, ?)

5

2

4

1

4

3

Section 13-3

Parallel and Perpendicular Lines


1-20

Excluding 14

Objectives

A. Understand the relationship

between the geometric and

algebraic definitions of parallel

and perpendicular lines.

B. Determine if 2 lines are parallel,

perpendicular, or neither.

C. Find the slopes of lines parallel or

perpendicular to a given line.

Geometric versus Algebraic ‘speak’

The number 1 lesson from this chapter is to show the tight linkage between geometry and algebra.

One key to understanding this ‘linkage’ is to understand the different ‘dialects’ we speak in the mathematical language.

Relate this to the many ‘dialects’ within the United States. The vast majority of citizens of the United States speak an English ‘dialect’. In other words, the roots of our language are the same whether we are from the deep south or from the industrial northeast. However, the words we use and the way we pronounce them are different, thus creating the ‘dialect’. We mean the same thing, but say the things differently.

Geometric versus Algebraic ‘speak’ - continued

•When we speak

‘geometrically’, we use terms

such as:

–Points

–Lines

–Planes

–Angles

–Polygons

–Solids

•When we speak

‘algebraically”, we use terms

such as:

–Numbers

–Variables

–Equations

–Sets

–Inequalities

Both the geometric and algebraic ‘dialects’ are a part

of the language of mathematics.

Parallel Lines

From Chapter 3, the geometric definition of parallel lines is:

Two coplanar lines that do not intersect.

Theorem 13-3 give the algebraic definition of parallel

lines:

Two non-vertical lines are parallel if and only if their

slopes are equal.

In other words, if m1=m2, then line 1 is parallel to line 2.

Perpendicular Lines

From Chapter 3, the geometric definition of perpendicular

lines is:

Two coplanar lines that intersect at right angles.

Theorem 13-4 give the algebraic definition of

perpendicular lines:

Two non-vertical lines are perpendicular if and only if the

product of their slopes is (- 1).

In other words, if (m1 x m2 = -1), then line 1 is

perpendicular to line 2.

Sample Problems

1. Find the slope of (a) AB (b) any line parallel to AB (c) any line

perpendicular to AB. A(- 2, 0) & B(4, 4)

1 2 3

y

x

1

2

3

A (-2, 0)

B (4, 4)

Label the points: A = P1 = (-2, 0); B = P2 = (4, 4)


x1 = -2, y1 = 0, x2 = 4, y2 = 4

2 1

2 1

riseslope m

run

y yy

x x x


2 1

2 1

4 0 4 2slope

4 2 6 3

y y

x x

Two non-vertical lines are parallel if and

only if their slopes are equal. 2

slope of parallel line3

Slope of a parallel line?

Sample Problems

1. Find the slope of (a) AB (b) any line parallel to AB (c) any line

perpendicular to AB. A(- 2, 0) & B(4, 4)

1 2 3

y

x

1

2

3

A (-2, 0)

B (4, 4)

Label the points: A = P1 = (-2, 0); B = P2 = (4, 4)


x1 = -2, y1 = 0, x2 = 4, y2 = 4

2 1

2 1

riseslope m

run

y yy

x x x


2 1

2 1

4 0 4 2slope

4 2 6 3

y y

x x

Slope of a perpendicular line?

3slope of perpendicular line

2

Two non-vertical lines are perpendicular if

and only if the product of their slopes is (- 1).

1 2 1m m 2

21

3m

Sample Problems

3. OEFG is a parallelogram.

What is the slope of each side?

O

E(2, 7) F

G

What else do you know?

What are the coordinates of point O? O (0, 0)


x1 = 0, y1 = 0, x2 = 2, y2 = 7

2 1

2 1

riseslope m

run

y yy

x x x

7 0 7slope EO

2 0 2

slope EO ? slope FG ?

FG || so?EO

7slope FG

2

slope OG ? slope OG 0 slope EF ? slope EF 0

Sample Problems

5. What is the slope of LM & PN?

Why are they parallel? What is

the slope of MN & LP? Why are

they not parallel? What kind of

quadrilateral is LMNP?

M(- 4, 2)

N(2, 4)

P(4, - 2) L(- 3, - 1) slope LM ?

2 1 3slope LM 3

4 3 1

slope PN ?4 2 6

slope PN 32 4 2

|| PN why?LM

slope MN ?4 2 2 1

slope MN2 4 6 3

slope LP ?2 1 1

slope LP4 3 7

not parallel to MN why?MN

What kind of quadrilateral is LMNP? Trapezoid

Sample Problems

7. Find the slope of each side and each altitude of ABC.

A(0, 0) B(7, 3) C(2, - 5) What should you do first?

1 2 3

y

x

1

2

3

A (0, 0)

B (7, 3)

C (2, -5)

slope AB ?3 0 3

slope AB7 0 7

slope AC ?5 0 5

slope AC2 0 2

slope CB ?3 5 8

slope CB7 2 5

What must be true of the altitude

drawn from the line containing AB?

Altitude must be to

the line containing AB.

slope AB slope of altitude = -13

slope of altitude = -17

7slope of altitude from AB = -

3

slope of altitude

from AC = ? 2

5

slope of altitude

from CB = ? 5

8

Sample Problems

9. Identify the legs of right RST. R(- 3, - 4) S(2, 2) T(14, - 8)

11. Given parallelogram ABCD, A(- 6, - 4) B(4, 2) C(6, 8)

D(- 4, 2). Show that opposite sides are parallel and opposite

sides are congruent.

13. R(- 4, 5) S(- 1, 9) T(7, 3) U(4, - 1) Show that RSTU is a

rectangle. Show that the diagonals are congruent.

Decide what type of quadrilateral HIJK is, then tell why.

15. H(0, 0) I(5, 0) J(7, 9) K(1, 9)

17. H(7, 5) I(8, 3) J(0, - 1) K(- 1, 1)

19. Point N(3, - 4) is on the circle x2 + y2 = 25. Find the slope

of the line tangent to the circle at N.

Section 13-4

Vectors


1-30

Excluding 8, 16

Objectives

A. Understand and utilize the terms

‘vector’, ‘magnitude’, and

‘direction’.

B. Identify and calculate the

horizontal and vertical

components of a vector.

C. Recognize and identify equal, parallel, and perpendicular

vectors.

D. Calculate the magnitude and slope of a vector.

E. Perform vector addition, vector subtraction, and scalar

multiplication.

vector: a line segment with both magnitude and direction,

written as an ordered pair of numbers (H, V) where H is

the horizontal component and V is the vertical component.

Vectors

x

y

H

V

horizontal component: the distance traveled left or right

from the starting to the ending point, found by taking

xstop - xstart

vertical component: the distance traveled up or down from

the starting point to the ending point, found by taking

ystop - ystart

Vector Components

(3, 5)

(7, 8)

H = Stop – Start = 7 – 3 = 4

V = Stop – Start

V = 8 – 5 = 3

Vector (H, V) = (4, 3)

Vector Terms

• Equal Vectors vectors with the same magnitude and

direction

– Note that equal vectors may or may not be collinear

• Magnitude of a Vector The length of the vector.

– Found by using the distance formula

• dot product: (H1, V1) (H2,V2) = H1H2 + V1V2

vector addition: (H1, V1) + (H2,V2) = (H1 + H2 , V1 + V2)

vector subtraction: (H1, V1) - (H2,V2) = (H1 - H2 , V1 - V2)

scalar multiplication: k(H1,V1) = (kH1, kV1)

22 VHAB :magnitude

H

V :slope

Sample Problems

Find AB and

3. A(6, 1) B(4, 3)

AB

‘Order’ of vector dictates that Point A is the

start point and Point B is the stop point.

1 2 3

y

x

1

2

3

A (6, 1)

B (4, 3)


H = xstop – xstart = 4 – 6 = -2

V = ystop – ystart = 3 – 1 = 2

( , ) ( 2,2)H V AB

?AB

( , )H VAB

?AB

2 2 AB H V

2 2( 2) (2) 4 4 2 2 units AB

Sample Problems

Perform the scalar multiplication.

11. 3(4, - 1)

The scalar k is 3. The horizontal component H is 4.

The vertical component V is -1.

k(H,V) = (kH, kV)

3(4, -1) = (3 x 4, 3 x -1) = (12, -3)

Sample Problems

Find the vector sum.

21. (3, - 5) + (4, 5)

1 1 1

2 2 2

, 3, 5

, 4,5

u H V

u H V

3 1 2 1 1 2 2, ,u u u H V H V

3 1 2 ?u u u

3 1 2 1 1 2 2 1 2 1 2, , , 3 4, 5 5 1, 10u u u H V H V H H V V

(H1, V1) + (H2, V2) = ?

(H1, V1) + (H2, V2) = (H1 + H2 , V1 + V2)

3 1 2 1 1 2 2 1 2 1 2, , , 3 4, 5 5 7,0u u u H V H V H H V V

Sample Problems – Another look at #21!


21. (3, - 5) + (4, 5)

1 2 3

y

x

1

2

3

C (7, 0)

A(0, 0)

(3, 5) AB

(3, 5) (4,5) AC

(4,5)BC

AC AB BC

(3, 5) (4,5) (3 4, 5 5) (7,0) AC

B (3, -5)

Sample Problems

27. An object, K, is being pulled by two forces KX = (- 1, 5) and

KY = (7, 3). What single force has the same effect as the two forces

acting together? What is the magnitude of this force?

1 2 3

y

x

1

2

3

K(0, 0)

The result of 2 forces working on an

object at the same time has the SAME

result as the object being worked on by

the first force and THEN being worked

on by the second force.

KX = (- 1, 5)

KY = (7, 3)

R(6, 8)

KR KX KY

( 1,5) (7,3) (6,8) KR

2 2KR H V

6 8 36 64 100 10 units 2 2KR

X(-1, 5)

Y(6, 8)

Sample Problems

Find AB and

1. A(1, 1) B(5, 4)

5. A(3, 5) B(- 1, 7)

7. A(0, 0) B(5, - 9)

9. A(- 1, - 1) B(- 4, - 7)

Perform the scalar multiplication.

13.

15.

AB

9 ,63

1

4)- ,6(2

1

Sample Problems

17. The vectors (8, 6) and (12, k) are parallel. Find k.

19. The vectors (8, k) and (9, 6) are perpendicular. Find k.


23. (- 3, - 3) + (7, 7)

25. (7, 2) + 3(- 1, 0)

Sample Problems

29. M is the midpoint of AB. T is the trisector of AB.

A(2, 3) B(20, 21). Find AB, AM and AT. Find the

coordinates of M & T.

A(2, 3)

B(20, 21)

M

T

Section 13-5

The Midpoint Formula


1-20

Excluding 18

Objectives

A. Understand and utilize the

midpoint formula.

Theorem 13-5 (Midpoint Formula)

The midpoint (xm , ym) of a segment that joins the points

(x1 , y1) and (x2 , y2) is the point

.2

yy,

2

x+xy,x 2121

mm

(x1 , y1) (xm , ym) (x2 , y2)

Sample Problems

Find the coordinates of the midpoint.

3. (6, - 7) & (- 6, 3)

1 2 1 2midpoint ,2 2

x x y y

1 2

1 1 2 2

6, 7 6,3

6, 7 6, 3

P P

x y x y

6 6 7 3

midpoint , 0, 22 2

midpoint ?P2 (-6, 3)

P1 (6, -7)

y

1 2 3 x -1

-2

-3


(0, -2)

Is this a REASONABLE answer?

Sample Problems

Find the length, slope and midpoint of PQ.

7. P(3, - 8) Q(- 5, 2)

Label the points: P1 = P = (3, -8), P2 = Q = (-5, 2)


x1 = 3, y1 = -8, x2 = -5, y2 = 2

2 2

length 5 3 2 8

64 100 2 41 units

2 8slope

5 3

10 5

8 4

midpoint ?


Q (-5, 2)

P (3, -8)

y

1 2 3 x -1

-2

-3

length distance = ?

2 2

2 1 2 1length x x y y

slope ?

2 1

2 1

slope y y

x x

1 2 1 2midpoint ,

2 2

x x y y

3 5 8 2

midpoint , 1, 32 2

Sample Problems

15. Find the midpoints of the legs and then the length of the

median of the trapezoid CDEF with vertices C(- 4, - 3),

D(- 1, 4), E(4, 4) & F(7, - 3) .

1 2 3

y

x

1

2

3

C(-4,-3)

D(-1,4) E(4,4)

F(7,-3)

,2 2

E F E FEF

x x y ymid

2

1,

2

5

2

1,

2

11


midpoint ,2 2

C D C DCD

x x y y

midpoint ?CD

4 1 3 4 5 1midpoint , ,

2 2 2 2CD

midpoint ?EF

4 7 4 3 11 1, ,

2 2 2 2EFmid

Sample Problems

15. Find the midpoints of the legs and then the length of the

median of the trapezoid CDEF with vertices C(- 4, - 3),

D(- 1, 4), E(4, 4) & F(7, - 3) .

1 2 3

y

x

1

2

3

C(-4,-3)

D(-1,4) E(4,4)

F(7,-3)

2

1,

2

5

2

1,

2

11

2 2

2 1 2 1distance x x y y

length distance = ?

2 2

22

11 5 1 1distance

2 2 2 2

160 8

2

Sample Problems

Find the coordinates of the midpoint.

1. (0, 2) & (6, 4)

5. (2.3, 3.7) & (1.5, - 2.9)

Find the length, slope and midpoint of PQ.

9. P(- 7, 11) Q(1, - 4)

M is the midpoint of AB. Find the coordinates of B.

11. A(1, - 3) M(5, 1)

13. A(0, 0) & B(8, 4), show P(2, 6) is on the perpendicular

bisector of AB.

Sample Problems

17. Show that OQ & PR have the same midpoint. What kind

of quadrilateral is OPQR? Show that the opposite sides

are parallel. Show the opposite sides are congruent.

O

P(2, 6) Q(9, 9)

R(7, 3)

Sample Problems

19. In right OAT M is the midpoint of AT. What are the

coordinates of M? Find MA, MT & MO. Find the

equation of the circle that circumscribes the triangle.

O

A(0, 8)

T(- 6, 0)

M

Section 13-6

Graphing Linear Equations


1-33 ALL

Excluding 1, 3, 32

Objectives

A. Identify linear equations.

B. Understand and utilize the

standard form of a linear equation.

C. Understand and utilize the slope-

intercept form of a linear

equation.

D. Understand and apply multiple methods for solving systems of

linear equations.

Remember!

A linear equation is any equation whose solution graph is a

line.

Methods for Graphing a Line:

1. Plot two or more points

• Remember any two points are contained on one unique line.

• For graphing purposes, I recommend you plot at least 3 points

1 2 3

y

x

1

2

3

12 xy

If x = ? Then y =

0 1

1 3

2 5

B

A slope

B

C,0point theisintercept -y

0,

A

Cpoint theisintercept -x

Theorem 13-6 (Standard Form of a Linear Equation)

The graph of any equation that can be written in form Ax + By = C

where A is zero or positive, A and B are not both zero, and A, B

and C are integers, is a line. Therefore, Ax + By = C is the

standard form of a linear equation.

Theorem 13-7 (Slope Intercept Form of a Line)

A line with the equation y = mx + b has slope m and

y-intercept b.

Methods for Graphing a Line:

1. Plot two points

2. Plot one point (Y-intercept) and rise and run according to the slope.

1 2 3

y

x

1

2

3

12 xy

y = mx + b

Slope-intercept

form of a linear

equation is?

1

2or 2 mm

b = 1

y-intercept = (0, 1)

Solving Systems of Linear Equations - Graphing

• Graph the two linear equations and identify the intersection

point.

– The point of intersection of the two lines gives the x and

y coordinates that will make BOTH linear equations

true.

– Can be used on any system but your answer is only as

accurate as your graph.

Solving Systems of Linear Equations - Graphing

1 2 3

y

x

1

2

3

1

1

xy

xy

Point of intersection (-1, 0)

01)1(1

01)1(1

answer!your CHECK

xy

xy

Solving Systems of Linear Equations –

Isolate and Substitute

• Solve for (isolate) x or y in one equation.

• Substitute the expression from the step above into the

second equation.

• Solve the second equation for the remaining variable.

• Substitute the answer from the step above into either

original equation and solve for the other variable.

• Can be used on any system but works best when one

coefficient is either 1 or - 1.


Isolate and Substitute

1723

122

yx

yx

Isolate (solve for) y in the first equation y = 12 – 2x

Using the result of the isolation, substitute for y in the

second equation 3x – 2(12 – 2x) = -17.

Solve for x 3x – 24 + 4x = -17

7x = 7 x = 1

Substitute x value back into original equation

2(1) + y = 12 y = 10

Solution (1, 10)


Linear Combination

• Addition: add the two equations to eliminate one of the

variables. Works only if the coefficients of one of the

variables are opposites.

• Subtraction: subtract the two equations to eliminate one of

the variables. Works only if the coefficients of one of the

variables are the same.

• Multiply one or both equations by a constant in order to

create coefficients of the same variable that are either the

same or opposites, then add/subtract.


Linear Combination

643

1625

yx

yx

What to do?

Graphing would be complicated.

Isolating x or y would leave nasty fractions.

Adding or subtracting equations will not eliminate variable.

Multiply both sides of first equation by 2 10x + 4y = 32

Now add the two equations:

2

26013

643

32410

x

yx

yx

yx

3

62

162)2(5

y

y

y Solution (2, 3)

Sample Problems

Find the x & y intercepts and the slope, then graph.

7. 3x + y = - 21 How do you find the x-intercepts for a solution graph?

Plug in zero for the y-value and solve for the x-value. 3x + 0 = -21

x = -7

How do you find the y-intercepts for a solution graph?

Plug in zero for the x-value and solve for the y-value.

x-intercept = (-7, 0)

3(0) + y = -21

y = -21 y-intercept = (0, -21)

How do you find the slope of the solution graph?

The equation 3x + y = -21 is in which form?

The equation 3x + y = -21 is in standard form.

What is the standard form of a linear equation?

Ax + By = C where A is positive or zero, A and B are not both zero,

and A, B, and C are integers.

Slope of the line when using standard form = ? A

mB

3

31

m

Any other way of finding the slope?

Convert the equation into slope-intercept form! y = -3x - 21

Sample Problems


7. 3x + y = - 21

Graph the solution.

x-intercept = (-7, 0) y-intercept = (0, -21) 3

31

m

(-7, 0)

(0, -21)

y

3 6 9 x -3

-6

-9

Sample Problems

Solve the system

29. 4x + 5y = - 7

2x - 3y = 13

What are the 3 methods you can use to solve systems of linear equations?

1. Graph 2. Isolate and substitute 3. Linear combination

Which of these would you use?

Why would you NOT use the graphing method for this problem?

Linear combination will be the method you use most frequently!

What would you do in order to do linear combination to this system?

Multiply this equation by 2 4x – 6y = 26

4x + 5y = - 7

- (4x - 6y = 26)

----------------- 0x + 11y = -33

y = -3

Plug back into equation!

4x + 5(-3) = - 7

4x – 15 = -7

4x = 8

x = 2

Solution (2, -3)

Sample Problems

1. Graph y = mx if m = 2, - 2,

3. Graph for b = 0, 2, - 2, - 4

5. Graph y = 0, y = 3, y = - 3


9. 3x + 2y = 12

11. 5x + 8y = 20

2

1- ,

2

1

bx2

1y

Sample Problems

Find the slope, x & y intercepts, then graph.

13. y = 2x - 3

15. y = - 4x

17.

19. 4x + y = 10

21. 5x - 2y = 10

23. x - 4y = 6

4x3

2y

Sample Problems

Solve the system

25. x + y = 3

x - y = - 1

27. x + 2y = 10

3x - 2y = 6

Section 13-7

Writing Linear Equations

Homework Page 555:

1-28

Excluding 6, 16

Final Answers must be in STANDARD FORM for a linear equation (Ax + By = C)

Objectives

A. Understand and utilize the point-

slope form of a linear equation.

B. Use various pieces of information

about a line or linear equation to

determine the standard, slope-

intercept, and/or point-slope form

of the linear equation.

Theorem 13-8 (Point Slope Form of a Line)

An equation of the line that passes through the point (x1, y1)

and has slope m is y - y1 = m(x - x1).

Writing an Equation of the Line

• Find the slope and one point on the line.

• Use the point-slope form of a line; put the slope in for m

and the point in for (x1 , y1).

• Distribute and rearrange the equation until it is in standard

form, with all coefficients being integers.

Sample Problems

Write the equation of the line in standard form.

5. slope = y-intercept (0, 8) 5

7

What is the point-slope form of a linear equation: 1 1y y m x x

Fill in the given point and slope: 7

8 05

y x

Put in standard form (Ax + By = C): 7

85

y x

71 8

5x y

7 5 40x y

Sample Problems

25. Write the standard form of the equation of a line through (5, 7)

and parallel to the line y = 3x – 4.

If the lines are to be parallel, then, m1 = m2. So m2 = 3.

You are given the point (5, 7).

What is the point-slope formula?

7 3 15

3 8

y x

x y

What will be the slope of the ‘new’ line?

y = 3x – 4 is in what form?

What is the slope-intercept form of a linear equation?

y = mx + b where m is the slope and b is the y-intercept.

What is be the slope of the line y = 3x - 4? m1 = 3

1 1y y m x x

Fill in what you know. 7 3 5y x Put in standard form.

Sample Problems

Write the equation of the line in standard form.

1. slope = 2 y-intercept = (0, 5)

3. slope = y-intercept (0, - 8)

5. slope = y-intercept (0, 8)

7. x-intercept (8, 0) y-intercept (0, 2)

9. x-intercept (- 8, 0) y-intercept (0, 4)

11. point (1, 2) slope = 5

13. point (- 3, 5) slope =

2

1

5

7

3

1

Sample Problems

15. point (- 4, 0) slope =

17. line through (1, 1) & (4, 7)

19. line through (- 3, 1) & (3, 3)

21. vertical line through (2, - 5)

23. line through (5, - 3) and parallel to the line x = 4

27. line through (- 3, - 2) and perpendicular to the line 8x - 5y = 0

29. perpendicular bisector of the segment joining (0, 0) & (10, 6)

31. the line through (5, 5) that makes a 45° angle measured

counterclockwise from the positive x axis

2

1

Section 13-8

Organizing Coordinate Proofs


1-10

Draw and label each diagram on

graph paper!

Objectives

A. Understand and apply the term

“coordinate proof”.

B. Given a polygon, choose an

appropriate placement of

coordinate axes on the polygon to

assist in a coordinate proof.

C. Given a polygon properly placed on a coordinate plane, choose

appropriate coordinates (variable and fixed) of the vertices of the

polygon to assist in a coordinate proof.

Coordinate Proofs

A coordinate proof:

– Is a method of proving a conditional statement

– Is similar to the direct and indirect proof methods you have already

encountered

– Has all of the requirements of any other proof method

• Identify the conditional statement to be proved

• Identify the given information and diagrams

• Identify the proof statement

• Provide logical steps from given information to proof

statement

– Can be used in conjunction with a direct or indirect proof.

– Uses objects placed in a coordinate plane to assist with the proof

– Uses variable coordinates from the coordinate plane to provide a

GENERAL proof of the conditional statement.

Coordinate Proof: Placing the Object on the Coordinate Plane

Consider placing an isosceles right triangle on a coordinate

plane. What are the characteristics of an isosceles right

triangle?

– Polygon with 3 sides

– Has exactly one right angle

– Has congruent legs


Placing an isosceles right triangle on a coordinate plane.

y

x 1

2

3

1 2 3

Does this appear to be a good

placement of the triangle in the

coordinate plane? Why or why not?

Is it easy to prove there is a right

angle in the triangle?

Is it easy to prove there are congruent

legs in the triangle?

Are the coordinates of the vertices of

this triangle fixed real numbers?

If we wanted to generalize the proof,

how many variable coordinate would

be required?

Would calculation of the slopes of the

lines be an easy task?

Would calculation of the lengths of the

sides be an easy task?



y

x 1

2

3

1 2 3

Can we make our life easier? Is there any rule that disallows us

from rotating the triangle on the

coordinate plane?

NO! So choose wisely!



y

x

Is it easy to prove there is a right

angle in the triangle?

What is the x-coordinate of this point?

(0, ?)

What is the y-coordinate of this point?

Be sure that it is useful in a general

proof!

(0, a)



Be sure that it is useful in a general

proof and it makes an isosceles

triangle!

(?, 0) (a, 0)



(0, ?) (0, 0)



y

x

Is it easy to prove there are congruent

legs in the triangle?

(0, a)

(a, 0)

Are the coordinates of the vertices of

this triangle fixed real numbers?

How many variable coordinate would

be required?

Would calculation of the slopes of the

lines be an easy task?

Would calculation of the lengths of the

sides be an easy task?

(0, 0)


Steps to properly placing a figure on a coordinate plane to

assist in a coordinate proof:

1. If one or more right angles exist in the figure, place one of

them at the intersection of the coordinate axes (origin).

2. If one or more sets of parallel lines exist in the figure,

place at least one of the parallel sides on either the x-axis

or the y-axis.

3. In MOST figures, it is best to use the origin as one of the

vertices of the figure.

4. Whenever possible, place other vertices on the

x-axis (x, 0) or on the y-axis (0, y).


Some Sample Diagrams

scalene triangle

scalene triangle scalene right triangle

(?, ?) (0, 0) (?, ?) (a, 0)

(?, ?) (0, b)

(?, ?) (?, ?) (a, 0)

(?, ?) (b, c)

(?, ?)

(0, 0)

(a, 0)

(?, ?) (0, b)

(?, ?) (c, 0)


rectangle

isosceles triangle

(?, ?) (a, 0) (?, ?) (-a, 0)

(?, ?) (0, b)

(?, ?) (0, 0)

(?, ?) (a, b)

(?, ?) (2a, 0)

(?, ?) (0, 0)

(?, ?) (0, a)

(?, ?) (b, 0)

(?, ?) (b, a)

(?, ?) (0, 0) (?, ?) (a, 0)

(?, ?) (b, c) (?, ?) (a + b, c)

isosceles triangle

parallelogram


isosceles trapezoid

(?, ?) (0, 0) (?, ?) (a, 0)

(?, ?) (b, c) (?, ?) (a - b, c)

trapezoid

(?, ?) (0, 0) (?, ?) (a, 0)

(?, ?) (b, c) (?, ?) (d, c)

Sample Problems

Supply the missing coordinates without using any new variables.

(a, b)

(?, ?)

(?, ?)

1. rectangle

(f, 0) (- f, 0)

(?, ?) (?, ?)

3. square

(m, n)

(h, 0)

(?, ?)

5. parallelogram

(?, ?)

(s, 0)

7. equilateral triangle

Sample Problems

(c, 0)

(?, b) (?, ?)

9. rhombus

Section 13-9

Coordinate Geometry Proofs

Homework Page 562:

1-10

Make sure you do two-column proofs and include a properly annotated

diagram!

Objectives

A. Prove conditional statements by

using coordinate geometry proof

methods.

Writing Coordinate Proofs

• To write a coordinate proof of a geometric theorem the

first step is to place the diagram on a generic graph i.e. a

graph without numbers.

• The second step is to label each vertex with variable

coordinates. Each x-coordinate and each y-coordinate

must be assigned a different variable unless a relationship

has already been proven to exist.

– To minimize the number of letters used to label the

vertices, it is wise to place one vertex on the origin and

as many other vertices as possible on the coordinate

axes.

Writing Coordinate Proofs

• The third step is to write the given and the proof statements

as algebraic expressions, using the coordinates from the

diagram.

• The fourth step is to create the body of the proof by using

the rules of algebra to transform the given equation into the

equation in the prove statement.

– Write as two-column proofs.

– Most proofs can be accomplished by using the

equations from theorems 13-1 to 13-7 along with the

definitions of slope and vectors to write or transform

your given statement.

Coordinate Geometry Proofs - Example

Prove that the diagonals of an isosceles trapezoid are congruent.

First step? Diagram on a generic graph (without numbers).

R (0, 0) U (n, 0)

S (p, q) T (n - p, q) The second step is to label each vertex with

variable coordinates. Each x-coordinate and

each y-coordinate must be assigned a different

variable unless a relationship has already been

proven to exist.

Second step?

The third step is to write the given and the proof statements as algebraic

expressions, using the coordinates from the diagram.

Third step?

How would you show these diagonals to be congruent? Distance formula?

2 2

2 1 2 1d x x y y 2 2 2 20 0RT n p q n p q

2 2 2 20US n p q n p q

Since RT = US, the diagonals are congruent.

y

x



The fourth step is to create the body of the

proof by using the rules of algebra to

transform the given equation into the equation

in the prove statement.

Fourth step?

R (0, 0) U (n, 0)

S (p, q) T (n - p, q)

y

x

2 2

2 1 2 12. d x x y y

1. Given 1. Quadrilateral RSTU; ST || RU;

RS TU; diagram with coordinates given.

1 1 2 2

2 2

2 1 2 1

2. The distance d between

points (x , y ) and (x , y ) is given by

d x x y y



The fourth step is to create the body of the

proof by using the rules of algebra to

transform the given equation into the equation

in the prove statement.

Fourth step?

R (0, 0) U (n, 0)

S (p, q) T (n - p, q)

y

x

2 2 2 23. 0 0RT n p q n p q

2 2 2 24. 0US n p q n p q

3. Substitution and simple math.

4. Substitution and simple math.

5.US RT

6.US RT 6. Definition of congruence

5. Substitution.

Sample Problems

Prove that the segment joining the midpoints of the legs of a trapezoid is parallel to the bases and has a length equal to half the sum of the lengths of the bases.

y

x O P (a, 0)

N (b, c) M (d, c)

E F

What might be useful to add to the diagram?

What is the midpoint formula?

ON

0 0midpoint , ,

2 2 2 2

b c b cE

MP

0midpoint , ,

2 2 2 2

d a c d a cF

1 2 1 2midpoint ,2 2

x x y y

What are the coordinates of point E?

What are the coordinates of point F?

Sample Problems

Prove that the segment joining the midpoints of the legs of a trapezoid is parallel to the bases and has a length equal to half the sum of the lengths of the bases.

y

x O P (a, 0)

N (b, c) M (d, c)

E F

,2 2

b cE

,2 2

d a cF

How would you prove EF || OP || MN?

2 1

2 1

y ym

x x

?EF

m 02 2 0

2 2 2

EF

c c

md a b d a b

?OP

m 0 0 0

00OP

ma a

What is the slope formula?

?MN

m

00

MN

c cm

d b d b

Sample Problems

Use coordinate geometry to prove each statement. First draw a figure and

choose convenient axes and coordinates.

1. The diagonals of a rectangle are congruent. (Theorem 5-12)

Sample Problems

Use coordinate geometry to prove each statement. First draw a figure and

choose convenient axes and coordinates.

3. The diagonals of a rhombus are perpendicular. (Theorem 5-13) (Hint:

Let the vertices be (0, 0), (a, 0), (a + b, c), and (b, c).

Show that c2 = a2 – b2.)

Sample Problems

5. Prove that the segment joining the midpoints of the diagonals of a trapezoid is parallel to the bases and has a length equal to half the difference of the lengths of the bases.

y

x O P (a, 0)

N (b, c) M (d, c)

Sample Problems

7. Prove that the figure formed by joining, in order, the midpoints of the

sides of quadrilateral ROST is a parallelogram.

y

x O S (2a, 0)

R (2b, 2c)

T (2d, 2e)

Sample Problems

9. Prove that the angle inscribed in a semicircle is a right angle. (Hint:

The coordinates of C must satisfy the equation of a circle.)

Chapter Thirteen

Coordinate Geometry

Review

Homework Page 568: 1-18

Homework Section 13-1 - St. Charles Preparatory School Links... · Coordinate Geometry USE GRAPH PAPER FOR HOMEWORK IN THIS CHAPTER, ... lines, and circles on a ... 3 (-1, -4) (3,

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