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Unit 5 – Coordinate Methods

Lesson 1 PRACTICE PROBLEMS A Coordinate Model of a Plane

I can use a coordinate plane to represent and analyze

geometric properties of shapes.

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Investigation Practice Problem

Options Max Possible

Points Total Points

Earned Investigation 1: Representing

Geometric Ideas with Coordinates #1, 2, 3, 4, 5,

6 14 points

Investigation 2: Reasoning with Slopes and Lengths #7, 8, 9, 10 14 points

Investigation 3: Representing and Reasoning with Circles #11 8 points

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________/36 points !

** In order to earn credit for practice problems, ALL WORK must be shown.**

LESSON 1 • A Coordinate Model of a Plane 181

On Your Own

Applications

1 Use graph paper, the Line( command from the DRAW menu of your graphing calculator, or interactive geometry software to draw a model of a kite with vertices A(5, -6), B(7, -2), C(5, 2), and D(-9, -2).

a. Does your drawing appear to be that of a kite? Use careful reasoning with the coordinates to justify that ABCD is a kite, that is, a quadrilateral with exactly two pairs of congruent adjacent sides.

b. Draw the cross braces of the kite and find their lengths using coordinates.

c. Use coordinates to find the midpoints of −− AC and −− BD .

d. Justify that the midpoint of −− AC is on −− BD . How is this fact seen in your drawing?

2 Use graph paper, the Line( command from the DRAW menu of your graphing calculator, or interactive geometry software to draw a model of a school crossing sign. Assume the height is the same as the width of the base, and the length of the vertical edges is half that of the base. Locate the shape on the coordinate axes so that one side of the shape is on the x-axis and the y-axis is a line of symmetry.

a. Give the coordinates of each vertex.

b. Determine the length of each side using coordinates. Which pairs of sides are the same length?

c. Use coordinates to find the height of your model sign.

d. Find the area of your model sign.

3 The following program, designed for one type of graphing calculator, computes the distance between two points in a coordinate plane. The left-hand column is the program; the right-hand column describes the function of the commands.

DIST Program Program Function in ProgramClrHome Clears display screen

Input “X COORD”,AInput “Y COORD”,B Enters x- and y-coordinatesInput “X COORD”,C {of two pointsInput “Y COORD”,D

√((A–C)2+(B–D)2)→L Calculates distance and stores value in memory location L

Disp “DISTANCE IS”,L Outputs calculated distance with label

CPMP-Tools

INVESTIGATION 1

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to draw a kite with vertices A (5, -6), B (7, -2), C (5, 2), and D (-9, -2).

ALL WORK MUST BE SHOWN FOR CREDIT.

182 UNIT 3 • Coordinate Methods

On Your Own

a. Describe how this program uses the Distance Between Two Points Algorithm on page 167.

b. What does the program call the coordinates of the two points?

c. Explain how the processing portion actually calculates the distance.

d. Enter the program DIST in your calculator (modified as necessary for your particular calculator). Check your program for accuracy by testing several pairs of points.

4 Modify the program in Applications Task 3 so that it will compute the slope of a nonvertical segment determined by two points. Call your new program SLOPE. Check your program for accuracy by testing it with several points.

5 In Investigation 1, Problem 10 (page 168), you wrote a midpoint algorithm for calculating the coordinates of the midpoint of a segment. A program for a graphing calculator that will compute the midpoint of a segment is shown below.

MIDPT Program Program Function in ProgramClrHome 1. Clears display screenInput “X COORD”,A 2. ________________________Input “Y COORD”,B 3. ________________________Input “X COORD”,C 4. ________________________Input “Y COORD”,D 5. ________________________(A+C)/2→X 6. ________________________(B+D)/2→Y 7. ________________________Disp “MIDPOINT COORDS” 8. Displays words, MIDPOINT

COORDSDisp X 9. ________________________Disp Y 10. ________________________Stop 11. ________________________

a. Analyze this program and explain the purpose of each command line as was done for lines 1 and 8.

b. Enter the program MIDPT in your calculator. (Depending on your calculator, you may need to modify the commands slightly.) Test the program on pairs of points of your choosing.

6 Drilling teams from oil companies search around the world for new sites to place oil wells. Increasingly, oil reserves are being discovered in offshore waters. The Gulf Oil Company has drilled two high-capacity wells in the Gulf of Mexico 5 km and 9 km from shore, as shown in the diagram on page 183. The 20 km of shoreline is nearly straight, and the company wants to build a refinery on shore between the two wells. Since pipe and labor cost money, the company wants to find the location that will serve both wells and uses the least amount of pipe when it is laid in straight lines from each well to the refinery.

CPMP-Tools

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LESSON 1 • A Coordinate Model of a Plane 183

On Your Own

B C

A

DWell #2

Well #1

Shoreline

9 km

20 km

5 km

a. How can coordinates be used to model this situation?

b. What distance(s) should you try to minimize to use the least amount of pipe?

c. Do you think the refinery should be closer to B, to C, or at the midpoint of the shoreline? Make a conjecture.

d. Determine your best estimate for the location of the refinery. About how much pipe will be required?

e. There are several methods for solving this problem, including:

• Analyze tables or graphs of a function relating total length of pipe to distance of refinery from point B.

• Use point D, its reflection across ! "# BC , and congruent triangles.

• Use the click-and-drag feature of interactive geometry software.

Select a method different from what you used in Part d and use that method to solve this problem. Compare your answer with that found in Part d.

7 A CAD face-view drawing of a building includes a quadrilateral PQRSwhose vertices are given by the matrix:

P Q R S 4 8 14 10 4 -2 2 8

a. Sketch the quadrilateral on a coordinate grid.

b. What kind of quadrilateral is PQRS? Give reasons to support your response.

8 −−AB has endpoints A(-5, 0) and B(4, 3). −−CD has endpoints C(-3, 9) and D(1, -3). The equations of the lines containing −− AB and −− CD are x - 3y = -5 and 3x + y = 0, respectively.

a. How could you quickly check that these equations are correct?

b. Verify that the lines are perpendicular.

c. Find the point of intersection of −− AB and −− CD by solving the system of equations.

d. Find the midpoints of −− AB and −− CD . Compare your results with Part c.

e. What kind of quadrilateral is ACBD? Explain your reasoning.

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INVESTIGATION 2

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Explain why.

184 UNIT 3 • Coordinate Methods

On Your Own

9 In Check Your Understanding Part d (page 169), you discovered that the line segment connecting the midpoints of two sides of a particular triangle was parallel to the third side. You may have also noticed that the length of the midsegment was half the length of the third side. With coordinates, you can verify this is true for any triangle.

C(b, c)

A(0, 0) B(a, 0)

M(?, ?) N(?, ?)

y

x

Using the above placement of △ABC in a coordinate plane:

a. Find the coordinates of the midpoint M of −− AC . Of the midpoint N of −− BC .

b. Use coordinates to explain why −−− MN ∥ −− AB .

c. Show that MN = 1 _ 2 AB.

10 Quadrilateral ABCD is a rhombus with general coordinates.

a. Determine the coordinates of point C. D(b, c) C(?, ?)

A(0, 0) B(a, 0)

y

x

b. Show that −− AC ⊥ −− BD .

c. Show that −− AC bisects −− BD and that −− BD bisects −− AC .

d. Write a statement that summarizes this general property of rhombuses.

11 The circle shown below was produced from the DRAW menu of a graphing calculator using the command Circle(0,0,6). It is displayed in the standard ZSquare window.

a. Write an equation for the displayed circle.

i. Use your equation to find two points on the circle whose x-coordinate is 3.

ii. What two expressions could be used in the Y= menu to produce a graph of the circle?

INVESTIGATION 3

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LESSON 1 • A Coordinate Model of a Plane 185

On Your Own

b. Use the Circle( command to produce a circle with center at (2, 4) and radius 10. What might be a good window to use to display the circle?

i. Write an equation for the circle.

ii. Use your equation to find two points on the circle whose x-coordinate is 5.

c. Use the Circle( command to produce the circle defined by (x + 5)2 + (y - 8)2 = 84.

i. Write an equation for a circle that has the same center and is tangent to the x-axis.

ii. Write an equation for a circle that has the same center and is tangent to the y-axis.

iii. Write an equation for a circle that is tangent to both the x- and y-axes and is congruent to the given circle that you graphed. How many circles are possible? How are their centers related?

Connections

12 In Course 1, you may have conducted an experiment in which you placed several equal weights at various positions on a yardstick and found the balance point or center of gravity. The balance point corresponded to the mean of the distances from zero on the yardstick.

0 10 20 30 40 50 60

Test a similar idea for two-dimensional shapes.

a. Cut out a triangle from a sheet of cardboard or tag board that

is about the size of a 1 _ 2 -sheet of notepaper.

• Experiment with the cutout to try to find a point at which it will balance on the top of your finger or a pencil.

• Now place the cutout on a coordinate grid and record the coordinates of its vertices.

• Compute the mean of the x-coordinates and the mean of the y-coordinates. Locate this point on your coordinate grid and on the cardboard cutout.

• Verify by balancing that the point you found is the center of gravity.

b. Repeat Part a for a rectangle. For a parallelogram that is not a rectangle. What do you notice?

c. Repeat Part a for a quadrilateral that is not a parallelogram. What do you notice?

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