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Lesson 3B: Geometry and Graphing

 Lesson Outline:


1. Triangles, Angles, and Lines

2. Graphing

   a.      The Graph

   b.      Line Plotting – Linear Equation revisited

   c.      Distance between points

   d.      Midpoint between points


TRIANGLES, ANGLES AND LINES

Let’s say we have two parallel lines with another line passing through both of them. The corresponding angles will have the same degree measure.

If we add another line we create two triangles.  Two triangles with the same angles.

These are known as similar triangles with corresponding angles having the same degree measure, as indicated by the matching colors.

What’s more, each pair of corresponding sides form the same ratio.

 

Thus, you can calculate the length of an unknown side if you have the lengths of:

- A side corresponding to the unknown side

- Or two other corresponding sides

Note also that if you have two triangles and know that at least two sets of angles correspond (have the same measure), then the triangles are similar triangles.  The third pair of angles must be the same, because the three angles in all triangles add up to the same total (180º).

Problem:

What is x?

A)    12 km

B)     10 km

C)    8 km

D)    6.5 km

E)     There is not enough information

We know we have two similar triangles, because they have opposite angles and right angles in common.

  • So, we can set up a ratio with the lengths of corresponding sides to solve for x:

             x / 20   =  13 / 26

             x  =  (13 / 26) *  20

             = (1/2) * 20

             = 10 km        

            The answer is B

A right triangle is any triangle that has a vertex that is a right angle (90º).

The lengths of the sides of a right triangle always have a relationship expressed by:

a2 + b2= c2        where a and b are the “legs” an c is the

                         “hypotenuse.”

This is called the Pythagorean relationship

(named for Pythagoras who came up with it

long ago in Greece.)

 

Helpful tip:

If you have the lengths of any two sides of a right triangle, you can calculate the unknown third side.

Problem:

What is x in yards?

A)    2 yards

B)     6 yards

C)    18 yards

D)    21 yards

E)     36 yards

It is a right triangle so you can use the Pythagorean relationship:

a2 + b2 = c2     where the hypotenuse c = 10 and the leg a = 8 and the missing leg b = x

           8 2 + x2 = 102

                 64+ x2= 100

        x2  = 100 - 64

        x2 = 36

        x  = 6 feet

  • The answer asked for is in yards.  Remember 1 yd. = 3 ft.

           1 ft = 1/3 yds., 1/3 yard for very foot, so

            x = 6 feet * 1/3 yds. per ft. = 2 yards     

           The answer is A.

GRAPHING

In Algebra, we looked at the linear equation as a function with a set of ordered pairs.  Here we look at how the linear equation and its ordered pairs create a line on a coordinate plane or graph.

a. The Graph

The graph, or coordinate plane, has a horizontal x-axis and vertical y-axis, both of which are numbered.

 

Coordinates are a pair of numbers of the form (x, y) that can be graphed, or plotted, onto the graph at those x and y values.

 Example:

Plot the point (1, 2)

The x-coordinate (horizontal axis) is 1 and the y-coordinate (vertical axis) is 2.

Start at the center (0, 0), which is also called the origin.  First follow the x-axis to the right (positive), one unit.  Then move up 2 units (positive) and you arrive at (1, 2) on the graph.  Mark the point with a dot.

Note:  If you are asked to plot a point on the test, you will be provided with a coordinate grid.  You simply bubble in the correct spot on the grid.

b. Line plotting - Linear Equation revisited

Recall that a linear equation represents a line on a graph. 

The solutions for the equation create a line when plotted on a graph.

Example:

y = 0.5x + 1

We can plug in 2 for x to get a value for y.

         y = 0.5 (2) + 1

            = 1 + 1

            = 2                  

so one point on the line for this linear equation is (2,2).

You can plug in other values to create a set of ordered pairs.

x          y

3          2.5

2          2

1          1.5

0          1

-1         0.5

-2         0

-3         -0.5

This gives the coordinates for the line.  You can then plot the points given by the coordinates and draw a line that passes through them.

Note: On the test you will not be asked to plot an entire line as an answer.  However you may be asked about parts of this process or information that is revealed in a line plot.

Now that you understand what a line plot is, there is a simpler approach of getting this information out of a linear equation.

Recall that slope-intercept form gets its name because there is a slope, m, and y-intercept, b.

y = mx + b

The y-intercept is where the line crosses the y axis.  This is where x = 0.  So the simplest point to plot is    (0, b)

The slope m is a number that indicates the steepness of the line.  The bigger the number, the steeper the line is.  When it is positive, the line rises to the right (the positive direction on the x-axis).  When it is negative, the line rises toward the left.

 In the previous example,

y = 0.5x + 1

The slope is 0.5 and the y-intercept is 1.

We can use this information to plot a line.

1.      Start at (0,b).  In this case the y-intercept b = 1 so start at (0,1).

2.      Use the slope m to extend a line from the point.

This is done by treating the slope as a ratio.

slope = rise/run                  

- “rise” being the number of units up and “run” being the number of units over

   (to the left or right, depending on the sign)

In this case m = 0.5 or ½

rise       =   1

run       =    2

Up 1, over (to the right) 2.

If you try this, you will get the same line shown in the previous graph.

Continue to extend the line as you see fit, following the method using the ratio of the slope.

If you have a slope of m = -3, the ratio would be  

   _     3              Up 3 and

          1          over (to the left) 1        

c. Distance between points

The distance formula is included on the list of formulas under "Coordinate Geometry."

It is used to find the number of units between two points, given their coordinates.

This comes from the Pythagorean relationship.  Any line segment on a graph can be considered the hypotenuse of a right triangle. We find the lengths of the legs by subtracting the coordinates along each dimension:

In this case the distance between the two points is:

distance            = √( (3-1)2 + (3-1)2)

                        = √( (2)2 + (2)2)

                        = √(4 + 4)

                        = √(8)

d. Midpoint between points

The midpoint is a point halfway between two points.

When you have two points (x1,y1) and (x2,y2)

 

The midpoint formula simply takes an “average” or middle value of the x and y dimensions.  We will further discuss the concept of an “average” in the following and final math section on simple statistics concepts.

Click on the link below to move on to lesson 4.

Back: Math Lesson 3A | Next: Math Lesson 4


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