Bell Work:

Find the hypotenuse of a triangle with leg lengths of 5 and 6 cm.

Answer:

√61

Lesson 98:Distance Between Two Points, Slope Formula

In Lesson 97, we discussed the use of the Pythagorean theorem in algebraic form to find the missing side of a triangle. To find the length c of the triangle show below we write

c = 4 + 7

c = √65

2 2 2

c4

7

Note that we just took the positive square root when solving for c since c represents a length. In the next examples, we will only solve for the positive square root when solving for a length since lengths must be positive. The distance between two points is defined to be the length of the straight line segment joining the two given points. Therefore, if we are given the coordinates of two points, we can find the distance between the points by graphing the points, drawing the triangle, and then solving the triangle to find the length of the hypotenuse.

Example:

Find the distance between the points whose coordinates are (4, 2) and (-3, -2).

Answer:

Distance = √65

Example:

Find the distance between the points (3, -4) and (-5, 2).

Answer:

Distance = 10

In Lesson 75 the slope of a straight line was defined to be the ratio of the change in the y coordinate to the change in the x coordinate as we move from one point on the line to another point on the line. to demonstrate, we will find the slope of the line through the points (4, 3) and (-2, -2) by first graphing the points and drawing the line.

The sign of the slop of this line is positive because the graphed line segment points toward the upper right. We then find the rise over the run of the points. Thus the slope of the line is +5/6.

It is not necessary to graph the points to find the slope of the line. if we call the two points point 1 and point 2 and then the coordinates (x , y ) and (x , y ), respectively, we can derive a relationship from which the slop for the line through these points can be determined algebraically.

1 1 2 2

We label the legs of the right triangle formed y – y and x – x . It is important to note that these quantities may be negative as well as positive. The absolute values of these quantities are the lengths of the legs of the right triangle formed. The ration of the change in y coordinates, y – y , to the change in x coordinates, x – x , is defined to be the slope of the line passing through both points.

2 1 2 1

2 1

2 1

We summarize this as follows.

The slope of a line through two points is represented as

m = y – y

x – x

2 1

2 1

Example:

Find the slope of the line that passes through the points (-3, 4) and (5, -2).

Answer:

Slope = -3/4

Example:

Find the slope of the line that passes through the same points as the previous example but this time switch the number 1 and 2 points.

(5, -2) and (-3, 4)

Answer:

Slope = -3/4

HW: Lesson 98 #1-30