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Rectangular Coordinates and Graphs

Ordered Pairs

The Rectangular Coordinate System

The Distance Formula

The Midpoint formula

Graphing Equations

Ordered Pairs

An ordered pair consists of two components, written inside parentheses, in which the order of the components is important.

Example 1

Use the table to write ordered pairs to express the relationship between each type of entertainment and the amount spent on it.

Example 1

a. DVD rental sales

Solution:

Use the data in the second row:

(DVD rental sales, $71).

Example 1

b. Movie tickets

Solution:

Use the data in the last row:

(Movie tickets, $30).

The Rectangular Coordinate System

y-axis

Quadrant I

Quadrant II

P(a, b)

b

x-axis

a

0

Quadrant III

Quadrant IV

The Distance Formula

Using the coordinates of ordered pairs, we can find the distance between any two points in a plane.

y

Horizontal side of the triangle has length

Q(8, 3)

P(–4, 3)

x

R(8, –2)

Definition of distance.

The Distance Formula

y

By the Pythagorean theorem, the length of the remaining side of the triangle is

Q(8, 3)

P(–4, 3)

x

R(8, –2)

The Distance Formula

y

So the distance between (–4, 3) and (8, –4) is 13.

Q(8, 3)

P(–4, 3)

x

R(8, –2)

The Distance Formula

y

To obtain a general formula for the distance between two points in a coordinate plane, let P(x1, y1) and R(x2, y2) be any two distinct points in a plane. Complete a triangle by locating point Q with coordinates (x2, y1). The Pythagorean theorem gives the distance between P and R as

Q(x2, y1)

P(x1, y1)

x

R(x2, y2)

The Distance Formula

Note Suppose that P(x1, y1) and R(x2, y2) are two points in a coordinate plane. Then the distance between P and R, written d(P, R) is given by

Example 2

Find the distance between P(–8, 4) and

Q(3, –2.)

Solution:

Be careful when subtracting a negative number.

Using the Distance Formula

If the sides a, b, and c of a triangle satisfy a2 + b2 = c2, then the triangle is a right triangle with legs having lengths a and b and hypotenuse having length c?

DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE

Example 3Are points M(–2, 5), N(12, 3), and

Q(10, –11) the vertices of a right triangle.

Solution This triangle is a right triangle if the square of the length of the longest side equals the sum of the squares of the lengths of the other two sides.

DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE

Example 3y

N(12, 3)

M(–2, 5)

x

Q(10, –11)

DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE

Example 3y

N(12, 3)

M(–2, 5)

x

Q(10, –11)

DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE

Example 3y

N(12, 3)

M(–2, 5)

x

Q(10, –11)

DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE

Example 3y

N(12, 3)

M(–2, 5)

x

R(10, –11)

Colinear TRIANGLE

We can tell if three points are collinear, that is, if they lie on a straight line, using a similar procedure

Three points are collinear if the sum of the distances between two pairs of points is equal to the distance between the remaining pair of points.

DETERMINING WHETHER THREE POINTS ARE COLLINEAR TRIANGLE

Example 4Are the points (–1, 5), (2,–4), and (4, –10) collinear?

Solution The distance between (–1, 5) and (2,–4) is

DETERMINING WHETHER THREE POINTS ARE COLLINEAR TRIANGLE

Example 4Are the points (–1, 5), (2,–4), and (4, –10) collinear?

Solution The distance between (2,–4) and (4,–10) is

DETERMINING WHETHER THREE POINTS ARE COLLINEAR TRIANGLE

Example 4Are the points (–1, 5), (2,–4), and (4, –10) collinear?

Solution The distance between the remaining pair of points (–1, 5) and (4,–10) is

Midpoint Formula TRIANGLE

The midpoint of the line segment with endpoints (x1, y1) and (x2, y2) has coordinates

USING THE MIDPOINT FORMULA TRIANGLE

Example 5Use the midpoint formula to determine the following.

a. Find the coordinates of M of the segment with endpoints (8, –4) and (–6,1).

Solution

Substitute in the midpoint formula.

USING THE MIDPOINT FORMULA TRIANGLE

Example 5Use the midpoint formula to determine the following.

b. Find the coordinates of the other endpoint B of a segment with one endpoint A(–6, 12) and midpoint M(8, –2).

Solution

Let (x, y) represent the coordinates of B. Use the midpoint formula twice.

USING THE MIDPOINT FORMULA TRIANGLE

Example 5Let (x, y) represent the coordinates of B. Use the midpoint formula twice.

x-value of A

x-value of M

x-value of A

y-value of M

Substitute carefully.

The coordinates of endpoint B are (22, –16).

APPLYING THE MIDPOINT FORMULA TO DATA TRIANGLE

Example 6The graph indicates the increase of McDonald’s restaurants worldwide from 20,000 in 1996 to 31,000 in 2006.

Use the midpoint formula and the two given endpoints to estimate the number of restaurants in 2001, and compare it to the actual (rounded) figure of 30,000

APPLYING THE MIDPOINT FORMULA TO DATA TRIANGLE

Example 6Number of McDonald’s Restaurants

Worldwide (in thousands)

35

30

25

20

2006

1996

Year

APPLYING THE MIDPOINT FORMULA TO DATA TRIANGLE

Example 6Solution:2001 is halfway between 1996 and 2006. Find the coordinates of the midpoint of the segment that has endpoints (1996, 20) and (2006, 31).

Our estimate is 25,500, which is well below the actual figure of 30,000.

FINDING ORDERED PAIRS THAT ARE SOLUTIONS OF EQUATIONS TRIANGLE

Example 7For the equation, find at least three ordered pairs that are solutions

a.

Solution: Choose any real number for x or y and substitute in the equation to get the corresponding value of the other variable.

FINDING ORDERED PAIRS THAT ARE SOLUTIONS OF EQUATIONS TRIANGLE

Example 7a.

Solution:

Let x = –2.

Let y = 3.

Multiply.

Add 1.

Divide by 4.

This gives the ordered pairs (–2, –9) and (1, 3).

Find one more ordered pair that works!

FINDING ORDERED PAIRS THAT ARE SOLUTIONS OF EQUATIONS TRIANGLE

Example 7b.

Solution

Given equation

Let x = 1

Square both sides

One ordered pair is (1, 2).

Find two more ordered pair!

FINDING ORDERED PAIRS THAT ARE SOLUTIONS OF EQUATIONS TRIANGLE

Example 7c. A table provides an organization method for determining ordered pairs.

Solution:

Five ordered pairs are listed as solutions for this equation.

Graphing an Equation by Point Plotting TRIANGLE

Step 1 Find the intercepts.

Step 2 Find as many additional ordered pairs as needed.

Step 3 Plot the ordered pairs from Steps 1 and 2.

Step 4 Connect the points from Step 3 with a smooth line or curve.

GRAPHING EQUATIONS TRIANGLE

Example 8a. Graph the equation

Solution:

Step 1 Let y = 0 to find the x-intercept, and let x = 0 to find the y-intercept.

GRAPHING EQUATIONS TRIANGLE

Example 8a. Graph the equation

Solution:

Step 2 We find some other ordered pairs.

Let y = 3.

Let x = –2.

Multiply.

Add 1.

Divide by 4.

This gives the ordered pairs (–2, –9) and (1, 3).

GRAPHING EQUATIONS TRIANGLE

Example 8a. Graph the equation

Solution:

Step 3 Plot the four ordered pairs from Steps 1 and 2.

y

Step 4 Connect the points with a straight line.

x

GRAPHING EQUATIONS TRIANGLE

Example 8b. Graph the equation

Solution:

y

Plot the ordered pairs found in example 7b, and develop a fourth point to confirm the direction the curve will take as x increases.

x

GRAPHING EQUATIONS TRIANGLE

Example 8c. Graph the equation

Solution:

y

x

This curve is called a parabola.

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