Geometry 2.3 Ws Midpoint Assignment 10 9

If we want to find the distance between two points on a number line we use the distance formula:

$$AB=\left | b-a \right |\; or\; \left | a-b \right |$$


Example

Point A is on the coordinate 4 and point B is on the coordinate -1.

$$AB=\left | 4-(-1) \right |=\left | 4+1 \right |=\left | 5 \right |=5$$

If we want to find the distance between two points in a coordinate plane we use a different formula that is based on the Pythagorean Theorem were (x1,y1) and (x2,y2) are the coordinates and d marks the distance:

$$d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$$

The point that is exactly in the middle between two points is called the midpoint and is found by using one of the two following equations.

Method 1: For a number line with the coordinates a and b as endpoints:

$$midpoint=\frac{a+b}{2}$$

Method 2: If we are working in a coordinate plane where the endpoints has the coordinates (x1,y1) and (x2,y2) then the midpoint coordinates is found by using the following formula:

$$midpoint=\left ( \frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2} \right )$$


Video lesson

Find the midpoint of the line segment.

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Use a blackboard, whiteboard, or overhead projector to present the following information to the class.

Distance

“The distance between two points in a plane is measured by the length of the straight line that joins the points.” See the definition of the Distance Formula in the Vocabulary section. “If the points are defined as Cartesian coordinates, then the distance between them is represented by the formula, commonly called the Distance Formula:



where the two points are represented by the ordered pairs (x1, y1) and (x2, y2).”

“If we look at the Pythagorean Theorem, which expresses the length of the hypotenuse of a right triangle, c2 = a2 + b2, we can see the similarity between the two expressions. Taking the square root of both sides of the equation, . If we replace  with a and with b, then one looks exactly like the other.”

“Let’s work through an example of using the Distance Formula to find the length of the straight line that joins two ordered pairs, (−8, −3) and (7, 5). Note that for these two ordered pairs, x1 = −8, y1 = −3, x2 = 7, and y2 = 5, substituting these values of x and y into the Distance Formula,”

 

“Looking at the graph of the two ordered pairs (−8, −3) and (7, 5), the right triangle with vertices at (−8, −3), (7, −3), and (7, 5) has the line joining (−8, −3) and (7, 5) as its hypotenuse. From the graph, it’s easy to see that the length of the right triangle’s base is 15 [(−8, −3) to (7, −3)] and its other leg is the altitude from (7, −3) to (7, 5), which is 8. From the Pythagorean Theorem,

c2 = 152 + 82

c = , which is 17, the same result as from the Distance Formula.”

Midpoint

“From the graph of the hypotenuse, the line joining the two ordered pairs (−10, 0) and
(10, 2), it looks like the midpoint is close to the origin.”

“Is the midpoint of the base of the triangle near the y-axis?” (yes)

“Let’s find out where the midpoint is. From our definition of midpoint, the coordinates are .”

“Substitute x1, y1, x2, and y2.

 

The ordered pair of the midpoint is (, 1).”

Slope

“If the line determined by the points we have been working with, (−8, −3) and (7, 5), were the graph of a linear equation, where y is the dependent variable and x is the independent variable, how could we identify that equation?”

“Remember the slope-intercept form of a linear equation, y = mx + b; the y-intercept is b and the slope is m.”

“Now refer to the right triangle above. What is the y-intercept?” ( because , where m is the slope, (altitude of the right triangle divided by its height), and b is the y-intercept);

 

 

“To represent the slope, go back to the definition of slope in the Vocabulary section, ‘the tangent of the angle the line makes with the positive x-axis.’”

“In a right triangle, such as the one represented above, the tangent to the positive x-axis of the angle whose vertex is at (−8, −3) is the ratio of the opposite side of the right triangle (distance from (7, −3) to (7, 5)) to the adjacent side (distance from (−8, −3) to (7, −3)). That ratio is .”

“Go back to the point-slope form of the linear equation, y = mx + b; m =  and b = , so the equation is . The slope in this example is also seen on the path of the line whose distance we first measured. If we think about that point at (−8, −3) moving to the right and rising above the x-axis, we see that it travels 15 units to the right in the horizontal direction while traveling 8 units up in the vertical direction. The change in the y-direction is positive 8, and the change in the x-direction is positive 15. This is another way to describe the slope on a coordinate plane: change in y divided by change in x, and also represented as slope = .”

Part 1

Begin the next part of the lesson with a conversation about measurement. Ask students what they measure on a daily basis and what they use to get that measurement. Some examples might be time with a clock or a watch, amounts of ingredients with measuring spoons or cups, weight with a scale, and distance with an odometer. “What do we do if we don’t have our usual measuring tools?” Give them a few minutes to think about it.

Ask, “How many of you estimate your measurements? Or use a formula instead of the measuring tool? Today we are going to learn a formula that will help us measure distance if we didn’t have a measuring tape or yard stick.”

Use the Graphic Organizer (M-G-7-3_Lesson 3 Graphic Organizer.doc and M-G-7-3_Lesson 3 Graphic Organizer KEY.doc) for this activity. Have students fill out the organizer as well as do the examples. They can do the examples on their own and then pair up with a partner to discuss the examples.

Part 2: Ball Activity

Go to a big area such as the cafeteria or gym. Lay out two of the 8-foot ropes as if they are the x- and y-axes of the coordinate plane (the origin is at the middle of both ropes). As a demonstration, place two students anywhere in the “plane” and give them one of the athletic balls. “If [name of first student] passed the ball to [name of second student], how could we measure the distance of the pass? Keep in mind that I didn’t bring a measuring tape or yard stick.” Give students a few minutes to think about this and answer the question. (Answer: pick the points and label the ordered pairs of the coordinates.)

“Let’s label the places they are standing with coordinates. Let’s say [name of first student] is standing at (−2, −1) and [name of second student] is standing at (3, 4). How could I measure the pass now?” (Use the Distance Formula.)

Place students in groups of four. Lay out all the other 8-foot ropes the same way as the first set. Hand out the Ball Activity sheet (M-G-7-3_Ball Activity.doc). Students measure the distance of the pass, the midpoint between the two coordinates, and the slope of the line the ball rolls on from one point to the other. The directions are on the Ball Activity sheet.

If there are not enough ropes and balls, split the class in half. Half the class does the Ball Activity while the other half does the Dot Activity; then switch.

Part 3: Dot Activity

This activity can be done in pairs. Tape a sheet of poster-sized graph paper on the wall. Have students draw in the x- and y-axes. Hand out the Dot Activity sheet (M-G-7-3_Dot Activity.doc) as well as a sheet of sticky dots. [IS.6 - All Students]

 

Part 4

Hand out the Lesson 3 Exit Ticket (M-G-7-3_Lesson 3 Exit Ticket and KEY.doc) to evaluate whether students understand the concepts.

Extension:

  • Students can create a map of their community for someone new moving into the area. They should include the school, police and fire stations, gas stations, a hospital, and restaurants. The map should be drawn on the coordinate plane and each building should be labeled with the coordinates. Below the map, students should create an “Index” of the distances and midpoints between each of the establishments.

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