On Your Own. Applications. Unit 3. b. AC = 8 units y BD = 16 units C(5, 2)

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1 Applications 1 a. If students use uniform and y scales or ZSquare, they should have a shape that looks like a kite. Using the distance formula, DC = AD = 1 and BC = BA = 0. b. AC = 8 units y BD = 16 units C(5, ) c. The coordinates of the midpoint of AC are (5, -). The coordinates of the midpoint D(-9, -) B(7, -) of BD are (-1, -). A(5, -6) d. The equation for BD is y = -. Since the coordinates of the midpoint of AC are (5, -), it lies on BD. This is seen in the drawing by noticing that the midpoint of AC, B, and D are all on the same line. Student choices of coordinates may vary. One eample using a base of length 16 is provided. a. The coordinates are A(8, 0), B(8, 8), C y C(0, 16), D(-8, 8), and E(-8, 0). 1 b. Pairs of sides as shown in the photo are the same length. D B AE = 16, AB = DE = 8, BC = CD = 18 = 8 E c. The height is 16 since C has -8-4 coordinates (0, 16). d. The area of the sign is the sum of the areas of BCD and rectangle ABDE. Area of sign = 1_ (16)(8) + (16)(8) = 19 square units 8 4 A INSTRUCTIONAL NOTE Students should enter the program in their calculators as a time-saver since they will regularly need to find distances. It is also recommended, but not required, that students enter the slope and midpoint programs into their calculators. Some students have difficulty getting a program to run because of synta errors. Those students may need etra help by the teacher or a group member. Troubleshooting can be very educational. DIFFERENTIATION Problems 3, 4, and 5 have students create three separate programs. Challenge students to write one program that finds all three distance, slope, and midpoint. A Coordinate Model of a Plane T181

2 a. This program uses the algorithm by inputting the coordinates of the two points, calculating the distance, and outputting the result of the calculation. b. (A, B) and (C, D) c. The formula in the processing adds the differences squared of the -coordinates and y-coordinates and then takes the square root of that sum. d. Students should enter the program and verify that it runs. 4 SLOPE Program Program ClrHome Input X COORD,A Input Y COORD,B Input X COORD,C Input Y COORD,D Function in Program Clears display screen Enters - and y-coordinates of two points TECHNOLOGY TIP For TI calculators, when entering the commands ClrHome, Input, and Disp, do not simply type them in, rather insert the commands from the CATALOG (scroll down in the list until you find them). (D B)/(C A) M Calculates slope and stores value in memory location M Disp "SLOPE IS ",M Outputs calculated slope with label 5 a. Function in Program 1. Clears display screen. Stores the -coordinate of one point in A 3. Stores the y-coordinate of the same point in B 4. Stores the -coordinate of the second point in C 5. Stores the y-coordinate of the second point in D 6. Calculates the -coordinate of the midpoint and stores it in X 7. Calculates the y-coordinate of the midpoint and stores it in Y 8. Displays words, MIDPOINT COORDS 9. Displays the -coordinate of the midpoint 10. Displays the y-coordinate of the midpoint 11. Ends the program b. Students should enter the program and test it on pairs of points. A Coordinate Model of a Plane T18

3 6 a. Each point on the model may be represented by an ordered pair. One convenient choice is to locate B at the origin. b. The sum of the distances from y the refinery on the shoreline to the two oil wells should be minimized. c. The refinery should be closer to B. However, students conjectures may vary. d e. A reasonable estimate for the location of the refinery A(0, 5) E(, 0) D(0, 9) B Refinery C(0, 0) is at E(7, 0). This location means about 4.4 km of pipe would be required. The eact location, ( 50 _ 7, 0 ), requires about 4.1 km of pipe. Various methods are shown below. Analyze a table of y = (0 - ) + 9 or graph this function on the interval 0 0. Locate the smallest y value. Using grid paper or geometry software, reflect CD across BC. Then construct AD'. Find the location of the refinery by solving the linear equation y = - _ = 0 or by using geometry 0 software constructing the intersection of the -ais and AD'. TECHNOLOGY NOTE CPMP-Tools or other geometry software may be used in Parts d or e. If using CPMP-Tools, students will need to change the horizontal scale found under Options, Window Scale. Encourage them to turn on the coordinates under Options, Default Style Window. CPMP-Tools A Coordinate Model of a Plane T183

4 To use the click-and-drag feature of geometry software, construct the shape, construct a point E on BC, construct AE and DE. Measure the lengths of AE and DE. Calculate the sum AE + DE. Click and drag point E to find the minimum value for AE + DE. A range of values around = 7.14 will display the minimum pipe length of approimately 4.4 km. CPMP-Tools 7 a. y S(10, 8) P(4, 4) R(14, ) Q(8, -) b. PQRS is a square. Slope of PQ is - 3_. Slope of SR is - 3_. Slope of PS is _. Slope of QR is _. Since the consecutive sides have 3 3 slopes that are opposite reciprocals, the quadrilateral is either a rectangle or a square. PS = PQ = 5 7.1, so PQRS is a square. 8 a. A quick check would involve substituting the coordinates for each point into its respective equation to see if the point is on the line. b. These lines are perpendicular since the slopes are 1_ and -3, opposite 3 reciprocals of each other. c. Students might solve this system using matrices, substitution, or linear combinations. The point of intersection is (- 1_, _ 3 ). d. The midpoint of AB is (- 1_, _ 3 ). The midpoint of CD is (-1, 3). The midpoint of AB is the same as the intersection point of AB and CD. But the midpoint of CD is not the same point of intersection. e. Using the distance formula, AC = CB 9. and AD = BD 6.71; so by the definition, quadrilateral ACBD is a kite. T183A UNIT 3 Coordinate Methods

5 9 a. M ( b _, c, _ c ) ; N ( a + b ) b. The slope of MN is _ 0 a_ = 0 and the slope of AB is 0. So, the lines are parallel. c. MN = _ a + b AB = a So, MN = 1_ AB. 10 a. C(a + b, c) b. The slope of AC is - _ b = _ a c_ a + b. The slope of BD is _ -c a - b. The product of the slopes is c_ a + b _-c a - b = _-c a - b. Since b + c = a, _-c a - b = -c b + c - b = _ -c c = -1. y a D(b, c) C(a + b, c) A(0, 0) b B(a, 0) c Thus, the diagonals are perpendicular. c. The midpoint of AC is (_ a + b, _ c ). The midpoint of BD is (_ a + b, _ c ). Since this is the same point, the diagonals bisect each other. d. The diagonals of a rhombus are perpendicular bisectors of each other. 11 a. + y = 36 i. When = 3, y = 7. y = ± 7 = ±3 3 ii. y = ± 36 - A Coordinate Model of a Plane T184

6 b. Students should produce the circle using Circle(,4,10). One good window would be the standard ZSquare increased by a factor of. A window of Xmin = -30, Xma = 30, Xscl = 5, Ymin = -0, Yma = 0, Yscl = 5 works well. i. ( - ) + (y - 4) = 100 ii. When = 5, (y - 4) = 91, so y or y c. Students should use Circle (-5,8, (84)). i. ( + 5) + (y - 8) = 64 ii. ( + 5) + (y - 8) = 5 iii. There are four possibilities. Students should have one of the following equations. ( + 84 ) + (y + 84 ) = 84 ( - 84 ) + (y - 84 ) = 84 ( + 84 ) + (y - 84 ) = 84 ( - 84 ) + (y + 84 ) = 84 Relationships that could be described are: The centers are reflections across - or y-aes. The centers are rotations of 90, 180, or 70 of each other. The centers have ± 84 for each coordinate. The centers are vertices of a square centered at the origin with side lengths 84. Connections 1 a. Students should conduct the eperiment and verify that the center of gravity of a triangle with vertices A( 1, y 1 ), B(, y ) and C( 3, y 3 ) is the point with coordinates ( , y 1 + y + y ). b. Students should notice that in the case of a rectangle and a parallelogram that is not a rectangle, the center of gravity is the point of intersection of the diagonals. c. For quadrilaterals that are not parallelograms, the method outlined in Part a does not produce the center of gravity. INSTRUCTIONAL NOTE Task 1 requires students to have pieces of cardboard and grid paper. Attaching the grid paper to the triangle before cutting out the triangle allows students to cut out triangles with vertices that have integer coordinates. INSTRUCTIONAL NOTE Recall from Course 1 that parallelograms have half-turn symmetry about the point of intersection of the diagonals. Some students might be encouraged to show algebraically why the observation in Part b must be the case. A Coordinate Model of a Plane T185