164 UNIT 3 • Coordinate Methods

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Computer images on a screen are composed of lighted pixels (screen points) whose coordinates satisfy specific conditions. By specifying the conditions on pixels, CAD (computer-aided design) software and geometry drawing programs can be used to create all sorts of shapes and designs.

In this investigation, you will explore how a two-dimensional graphics program called Interactive Geometry uses coordinates in drawing and in calculating measures of geometric figures. Other software programs may work differently, but they are all based on the same mathematical ideas. As you work on the following problems, look for answers to these questions:

How can you create a polygon using interactive geometry software?

What information and calculations are needed to find slopes, lengths, and midpoints of sides?

Creating Shapes As a class or in pairs, experiment with the drawing capabilities of interactive geometry software.

1 Explore how each command in the Draw menu can be used to create examples of the objects listed.

a. Draw commands can also be implemented by selecting the appropriate tool displayed on the software toolbar.

Match each icon in your software toolbar with the corresponding Draw command.

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Name: _______________________________

Unit 5Lesson 1

Investigation 1

LESSON 1 • A Coordinate Model of a Plane 165

b. In a clear window, draw a circle congruent to the one on the computer screen shown on page 163. Explain how you know the two circles are congruent.

c. In a clear window, draw a quadrilateral congruent to the ones on the computer screen shown on page 163. Discuss how you know that the two quadrilaterals are congruent.

2 Now explore how to use interactive geometry software to create special quadrilaterals. First, clear the window.

a. Draw a rectangle. Record the coordinates of the vertices in the order in which you drew them. Discuss how you know that the displayed figure is a rectangle.

b. Clear the window and then find a different method to draw the same rectangle as in Part a. Describe your method.

c. Clear the window and then draw a parallelogram that is not a rectangle. Record the coordinates of the vertices in the order in which you drew them. Discuss how you know the displayed figure is a parallelogram.

d. By clicking and dragging a point, you can generate shapes for which some conditions remain the same and other conditions vary. Click and drag one of the vertices of your parallelogram in Part c. What types of shapes can you create?

Calculating Slopes and Lengths Next, explore some of the measurement capabilities of interactive geometry software.

3 Using a clear window, draw a rectangle ABCD with coordinates A(2, 9), B(10, 9), C(10, -6), and D(2, -6).

a. Draw the diagonals of the rectangle. b. Use the “Slopes” command in the

Measurements menu to find the slopes of the lines containing each side and the slopes of the two diagonals. Discuss why the reported slopes are reasonable.

c. How do you think the software calculates these slopes?

d. Calculate the slopes of the lines containing the diagonals without the use of technology. Compare your results to those in Part b. Explain any differences.

4 Delete the reported slopes. Then use the “Lengths” command in the Measurements menu to calculate the lengths of the sides of the rectangle you created in Problem 3.

a. How do you think the software calculates these lengths?b. Suppose you have two points with coordinates (a, b) and (c, b)

with a < c.

i. How do you know that the points are on a horizontal line?

ii. Write an algebraic expression for the length of the segment or distance between the points.

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Slope Formula:

166 UNIT 3 • Coordinate Methods

c. Suppose you have a vertical line.

i. What is true of the coordinates of all points on the line?

ii. Using variables, write coordinates for a point on the same vertical line as the point with coordinates (a, b).

iii. Write an algebraic expression for the length of the segment or distance between the points.

5 Next, use the “Lengths” command to find the lengths of the diagonals of the rectangle you created in Problem 3.

a. Explain how the software could use the coordinates of points A, C, and D to calculate the length of −− AC . How could the software use the coordinates of points A, C, and B to calculate the length of −− AC ?

b. Test your ideas in Part a by calculating the length of −− BD and compare your answer to the software calculation.

c. What theorem justifies the method you used?

d. Now consider points P(-1, 3) and Q(2, 7) in a coordinate plane.

i. Make a sketch on a coordinate grid showing points P and Q and −− PQ .

ii. Find the length of the segment −− PQ or distance between points P and Q. Compare your answer and method with those of your classmates. Resolve any differences.

e. Use similar reasoning to find the distance between points S(-5, 4) and T(3, -2).

6 To generalize the method you used for calculating distance between two points in a coordinate plane, consider general points A(a, b) and B(c, d) graphed below.

y

x

B(c, d)

A(a, b) C

a. Make a copy of the diagram showing the coordinates of point C.

b. Write expressions for the distances AC and BC.

c. Write a formula for calculating the distance AB. Compare your formula with that of your classmates and resolve any differences.

d. When points A and B are on a horizontal or vertical line, will your formula calculate the correct distance AB? Why?

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Distance Formula:

LESSON 1 • A Coordinate Model of a Plane 167

Interactive geometry software uses a method equivalent to yours to calculate the distance between two points. In order to do this, the software needs information or input (in this case, the coordinates of two points); instructions for processing the information (in this case, a formula); and then instructions on what to do with the results or output (in this case, it displays the distance). Specifying such instructions is called programming.

Before writing a program, it is helpful to prepare an algorithm that lists the main sequence of steps needed to accomplish the task. The Distance Between Two Points Algorithm below could be used to guide program writing for any computer or calculator. In Applications Task 3, you will analyze a calculator program that implements this algorithm.

Distance Between Two Points Algorithm Step 1: Input the coordinates of one point. Step 2: Input the coordinates of the other point. !

input

Step 3: Use the coordinates and the formula in Problem 6 Part c to calculate the desired distance. !

processing

Step 4: Display and label the distance. ! output

7 Use the questions below to help write a Slope Algorithm, similar to the Distance Between Two Points Algorithm, that could be used to prepare a program to calculate and display the slope of a line through the points A(a, b) and B(c, d).

• What information would you need to input?

• What formula could be used in the processing portion?

• What information should be displayed in the output?

Calculating Midpoints You now have a method for calculating the slope of a line and a method for calculating the distance between two points. Thus, you can compute the length and the slope of a segment in a coordinate plane. Coordinates also can be used by a graphics program to calculate the midpoint of a segment, that is, the point on a segment that is the same distance from each endpoint.

8 Use interactive geometry software to draw a rectangle ABCD with vertices A(2, 9), B(10, 9), C(10, -6), and D(2, -6).

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168 UNIT 3 • Coordinate Methods

a. Use the “Midpoint” command in the Construct menu (or select the corresponding icon in the toolbar) to find the midpoint of each side of the rectangle.

i. Record the coordinates of the midpoints.

ii. How do you think the software found these midpoints?

b. What should your software report as the midpoint of the segment with endpoints P(-3, 2) and Q(1, 2)? With endpoints S(3, 3) and T(3, -2)? Check your conjectures.

c. Suppose you have two points P(a, b) and Q(c, b). Write expressions for the coordinates of the midpoint of −− PQ . Compare your expressions with others and resolve any differences.

d. Repeat Part c for the case of points P(a, b) and Q(a, c).

9 Use the “Midpoint” command to find the midpoint of diagonal −− AC of the rectangle in Problem 8. Of diagonal −− BD . Record the coordinates.

a. What do you notice about the midpoints of the diagonals of the rectangle?

b. How do you think the software found these midpoints? Test your conjecture using a coordinate grid to find the midpoint of the segment joining the points S(-3, -3) and T(5, 5). Verify using your software.

c. Now consider a segment with general B(c, d)

A(a, b) C(c, b)

M

endpoints A(a, b) and B(c, d). Make a conjecture about the coordinates of the midpoint M of this segment.

i. Test your conjecture using a coordinate grid and the points V(2, -4) and W(6, 8).

ii. Check that your calculated point is the midpoint by verifying that:

• it is on the line containing the points V and W, and

• it is equidistant from points V and W.

10 Use the following questions to help write a Midpoint Algorithm that could be used to prepare a program to calculate and display the coordinates of the midpoint of a segment with endpoints A(a, b) and B(c, d).

• What information would you need to input?

• How will the processing portion of your algorithm differ from the algorithms to calculate distance and slope?

• What formula or formulas could be used in the processing portion?

• What information should be displayed in the output?

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Midpoint Formula: