Topic 7 Coordinate Geometry - Birdvilleschools
Topic 7 Coordinate Geometry TOPIC OVERVIEW VOCABULARY 7-1 Polygons in the Coordinate Plane 7-2 Applying Coordinate Geometry 7-3 Proofs Using Coordinate Geometry DIGITAL APPS English/Spanish Vocabulary Audio Online: English Spanish coordinate geometry, p. 296 geometría de coordenadas coordinate proof, p. 302 prueba de coordenadas PRINT and eBook Access Your Homework . . . Online homework You can do all of your homework online with built-in examples and “Show Me How” support! When you log in to your account, you’ll see the homework your teacher has assigned you. Your Digital Resources PearsonTEXAS.com Homework Tutor app Do your homework anywhere! You can access the Practice and Application Exercises, as well as Virtual Nerd tutorials, with this Homework Tutor app, available on any mobile device. STUDENT TEXT AND Homework Helper Access the Practice and Application Exercises that you are assigned for homework in the Student Text and Homework Helper, which is also available as an electronic book. 294 Topic 7 Coordinate Geometry
3--Act Math You Be the Judge Have you ever been a judge in a contest or competition? What criteria did you use to decide the winner? If you were one of many judges, did you all agree on who should win? Often there is a set of criteria that the judges use to help them score the performances of the contestants. Having criteria helps all of the judges be consistent regardless of the person they are rating. Think about this as you watch the 3-Act Math video. Scan page to see a video for this 3-Act Math Task. If You Need Help . . . Vocabulary Online You’ll find definitions of math terms in both English and Spanish. All of the terms have audio support. Learning Animations You can also access all of the stepped-out learning animations that you studied in class. Interactive Math tools These interactive math tools give you opportunities to explore in greater depth key concepts to help build understanding. Interactive exploration You’ll have access to a robust assortment of interactive explorations, including interactive concept explorations, dynamic activitites, and topiclevel exploration activities. Student Companion Refer to your notes and solutions in your Student Companion. Remember that your Student Companion is also available as an ACTIVebook accessible on any digital device. Virtual Nerd Not sure how to do some of the practice exercises? Check out the Virtual Nerd videos for stepped-out, multi-level instructional support. PearsonTEXAS.com 295
7-1 Polygons in the Coordinate Plane TEKS FOCUS VOCABULARY TEKS (2)(B) Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines. TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. Additional TEKS (1)(F), (1)(G), (6)(E) Coordinate geometry – the analytical use of algebra to study geometric properties of figures drawn on the coordinate plane Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING You can classify figures in the coordinate plane using the formulas for slope, distance, and midpoint. Key Concept Formulas and the Coordinate Plane Formula When to Use It Distance Formula To determine whether sides are congruent diagonals are congruent d 2(x2 - x1)2 (y2 - y1)2 Midpoint Formula x x y y M 1 2 2, 1 2 2 ( ) Slope Formula y2 - y1 m x -x 2 1 296 Lesson 7-1 Polygons in the Coordinate Plane To determine the coordinates of the midpoint of a side whether diagonals bisect each other To determine whether opposite sides are parallel diagonals are perpendicular sides are perpendicular
Problem 1 How do you classify a triangle? The terms scalene, isosceles, and equilateral have to do with the side lengths of a triangle. Use the Distance Formula to check whether any sides are congruent. TEKS Process Standard (1)(D) Classifying a Triangle Is ABC scalene, isosceles, or equilateral? y Use the Distance Formula to find the lengths of the sides. AB 2(4 - 0)2 (4 - 2 A 1)2 Simplify within parentheses. 116 9 x O Then simplify the powers. 125 5 B 4 The vertices of the triangle are A(0, 1), B(4, 4), and C(7, 0). 2 4 6 C Simplify. BC 2(7 - 4)2 (0 - 4)2 19 16 Simplify within parentheses. Then simplify the powers. 125 5 Simplify. 149 1 Then simplify the powers. CA 2(0 - 7)2 (1 - 0)2 150 512 hsm11gmse 0607 t06565.ai Simplify within parentheses. Simplify. Since AB BC 5, two sides of the triangle are congruent. By definition, ABC is isosceles. Problem 2 TEKS Process Standard (1)(G) Classifying a Quadrilateral How can you show that a quadrilateral is a rhombus? First, prove the quadrilateral is a parallelogram. Next, show that its diagonals are perpendicular. Prove ABCD is a rhombus. Step 1 Use the Slope Formula to verify the opposite sides are parallel. 6 y C B 4-0 slope of AB 4 - 1 - ( - 2) 1-5 slope of CD 2 - 3 4 5-4 slope of BC 1 3 - ( - 1) 4 1-0 slope of AD 1 2 - ( - 2) 4 x D A O 2 4 AB } CD and BC } AD, so ABCD is a parallelogram. Step 2 Use the Slope Formula to verify the diagonals are perpendicular. slope of AC 5-0 1 3 - ( - 2) slope of BD 1 - 4 2 - ( - 1) -1 The product of the slopes of the diagonals is -1, so AC # BD. Since quadrilateral ABCD is a parallelogram with perpendicular diagonals, it is a rhombus (Theorem 6-16). PearsonTEXAS.com 297
Problem 3 Besides using the Slope Formula to verify that opposite sides are parallel, how can you show that NPQR is a parallelogram? You can show it has two pairs of congruent opposite sides, or that its diagonals bisect each other. Verifying Parallelism of Line Segments Without using the Slope Formula, verify that NP } QR and PQ } NR. P N 4 y 2 x -6 Method 1 -4 -2 O R -2 Q2 Use the Distance Formula. First, find the distances between points N and P and between points R and Q to show that NP and QR are congruent. Then find the distances between points P and Q and between points R and N to show that PQ and NR are congruent. NP 2( -2 - ( -5))2 (3 - 2)2 PQ 2(1 - ( -2))2 (0 - 3)2 29 1 29 9 210 218, or 322 QR 2(1 - ( -2))2 (0 - ( -1))2 NR 2( -5 - ( -2))2 (2 - ( -1))2 29 1 29 9 210 218, or 322 Theorem 6–8 states if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. NP QR and PQ NR. Therefore, NPQR is a parallelogram. By the definition of a parallelogram, NP } QR and PQ } NR. Method 2 Use the Midpoint Formula. Find the midpoints of NQ and PR to determine whether they are the same point. ( - 52 1, 2 2 0 ) ( -2, 1) - 2 ( - 2) 3 ( - 1) midpoint of PR ( , ) ( -2, 1) 2 2 midpoint of NQ Theorem 6-11 states if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. The midpoints of NQ and PR are the same point, so the diagonals bisect each other. Therefore, NPQR is a parallelogram. By the definition of a parallelogram, NP } QR and PQ } NR. 298 Lesson 7-1 Polygons in the Coordinate Plane
Problem 4 Verifying Congruence of Segments How can you approach Problem 4 without using the Distance Formula? Find another way to verify that segments are congruent. Use the Perpendicular Bisector Theorem (Theorem 5-2). M is the midpoint of DE. Without using the Distance Formula, verify that FD FE. E Step 1 Use the Midpoint Formula to find the coordinates of M. M Step 2 Use the Slope Formula to show FM # DE. -6 7 3 -2 2 slope of FM 72 3 -1 - 2 - ( -2) -2 6-1 5 slope of DE 1 - 1 - ( - 6) 5 2 F D -4 6 4 M ( - 6 2( - 1), 1 2 6 ) ( - 72, 72 ) y x O -2 2 -2 The product of the slopes is -1, so FM # DE. NLINE HO ME RK O M is the midpoint of DE, so FM is the perpendicular bisector of DE. The Perpendicular Bisector Theorem (Theorem 5-2) states if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Since point F is on the perpendicular bisector of DE, then F is equidistant from the endpoints, D and E. Therefore, FD FE. WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Determine whether ABC is scalene, isosceles, or equilateral. Explain. For additional support when completing your homework, go to PearsonTEXAS.com. 1. 2. y A 3. y A y A 2 B x O C 2 2 C O 2 2 x B B x O 2 2 C 2 4. Explain Mathematical Ideas (1)(G) PQRS has vertices P( -4, 4), Q(2, 0), R(0, -3), S( -6, 1). Without using the Slope Formula, verify PQ } SR and QR } SP. Use the Distance Formula in your solution. hsm11gmse 0607 t06588.ai hsm11gmse 0607 t06589.ai hsm11gmse 0607 t06590.ai 5. Explain Mathematical Ideas (1)(G) PQRS has vertices P(8, 5), Q(5, -4), R( -1, -2), S(2, 7). Without using the Slope Formula, verify PQ } SR and QR } SP. Use the Midpoint Formula in your solution. 6. Explain Mathematical Ideas (1)(G) An isosceles triangle has vertices A(3, 3), B(8, 4), C(2, -2). M is the midpoint of BC. Without using the Distance Formula, verify that AB AC. Use the Perpendicular Bisector Theorem (Theorem 5-2) in your solution. PearsonTEXAS.com 299
Determine whether the parallelogram is a rhombus, rectangle, square, or none of these. Explain. 7. P( -1, 2), O(0, 0), S(4, 0), T(3, 2) 8. L(1, 2), M(3, 3), N(5, 2), P(3, 1) 9. R( -2, -3), S(4, 0), T(3, 2), V( -3, -1) 10. W( -3, 0), I(0, 3), N(3, 0), D(0, -3) 11. Apply Mathematics (1)(A) An artist is planning to paint a rectangle on a wall as part of a mural. Quadrilateral PQRS in the coordinate grid at the right represents the planned location of the rectangle. Is PQRS a rectangle? If so, explain your reasoning. If not, describe how the artist could change the plans to make sure PQRS is a rectangle. P S Q 12. Justify Mathematical Arguments (1)(G) A classmate says that if you can show that quadrilateral EFGH is a rhombus, then you only need to show that one pair of adjacent sides is perpendicular in order to prove that EFGH is a square. Is the classmate correct? Explain your reasoning. R Graph and label each triangle with the given vertices. Determine whether each triangle is scalene, isosceles, or equilateral. Then tell whether each triangle is a right triangle. 13. T(1, 1), R(3, 8), I(6, 4) 14. J( -5, 0), K(5, 8), L(4, -1) 15. A(3, 2), B( -10, 4), C( -5, -8) 16. H(1, -2), B( -1, 4), F(5, 6) 17. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Are the triangles at the right congruent? How do you know? y Q W 2 T 18. Display Mathematical Ideas (1)(G) A quadrilateral has opposite sides with equal slopes and consecutive sides with slopes that are negative reciprocals. What is the most precise classification of the quadrilateral? Explain. O P 2 x 6 R S Graph and label each quadrilateral with the given vertices. Then determine the most precise name for each quadrilateral. hsm11gmse 0607 t06585 19. P( -5, 0), Q( -3, 2), R(3, 2), S(5, 0) 20. S(0, 0), T(4, 0), U(3, 2), V( -1, 2) 21. F(0, 0), G(5, 5), H(8, 4), I(7, 1) 22. M( -14, 4), N(1, 6), P(3, -9), Q( -12, -11) 23. A(3, 5), B(7, 6), C(6, 2), D(2, 1) 24. N( -6, 4), P( -3, 1), Q(0, 2), R( -3, 5) 25. J(2, 1), K(5, 4), L(8, 1), M(2, -3) 26. H( -2, -3), I(4, 0), J(3, 2), K( -3, -1) 27. DE is a midsegment of ABC at the right. Show that the Triangle Midsegment Theorem (Theorem 5-1) holds true for ABC. 28. a. Describe two ways you can show whether a quadrilateral in the coordinate plane is a square. y B D 2 C E A O 2 x 4 6 b. Evaluate Reasonableness (1)(B) Which method is more efficient? Explain. 300 Lesson 7-1 Polygons in the Coordinate Plane hsm11gmse 0607 t06591
29. Apply Mathematics (1)(A) Interior designers often use grids to plan the placement of furniture in a room. The design at the right shows four chairs around a coffee table. The designer places cutouts of chairs on points where the gridlines intersect. She wants the chairs oriented at the vertices of a parallelogram. Does she need to fix her plan? If so, describe the change(s) she should make. 30. Connect Mathematical Ideas (1)(F) The diagonals of quadrilateral EFGH intersect at D( -1, 4). EFGH has vertices at E(2, 7) and F( -3, 5). What must be the coordinates of G and H to ensure that EFGH is a parallelogram? 31. Use the diagram at the right. A a. What is the most precise classification of ABCD? 4 D 33. 6 34. 10 35. n G x O 6 The endpoints of AB are A( 3, 5) and B(9, 15). Find the coordinates of the points that divide AB into the given number of congruent segments. 32. 4 F 2 b. What is the most precise classification of EFGH? c. Are ABCD and EFGH congruent? Explain. y 4 6 B C 4 E H hsm11gmse 0607 t06592 TEXAS Test Practice 36. In the diagram, lines / and m are parallel. What is the value of x? A. 5 C. 13 B. 12 D. 25 155 m (x 2 11) 37. K( -3, 0), I(0, 2), and T(3, 0) are three vertices of a kite. Which point could be the fourth vertex? F. E(0, 5) G. E(0, 0) H. E(0, -2) hsm11gmse 0607 t06593 J. E(0, -10) 38. In the diagram, which segment is shortest? A. PS C. PQ B. PR D. QR 39. A( -3, 1), B( -1, -2), and C(2, 1) are three vertices of quadrilateral ABCD. Could ABCD be a rectangle? Explain. P 57 Q 61 62 S 60 R PearsonTEXAS.com hsm11gmse 0607 t12944 301
7-2 Applying Coordinate Geometry TEKS FOCUS VOCABULARY TEKS Foundational to (2) The student uses the process skills to understand the connections between algebra and geometry and uses the one- and twodimensional coordinate systems to verify geometric conjectures. TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. Coordinate proof – In a coordinate proof, a figure is drawn on a coordinate plane and the formulas for slope, midpoint, and distance are used to prove properties of the figure. Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data. Additional TEKS (1)(F) ESSENTIAL UNDERSTANDING You can use variables to name the coordinates of a figure. This allows you to show that relationships are true for a general case. Key Concept Using Variables for Coordinates To place a general figure in the coordinate plane, it is usually helpful to place one side on an axis or to center the figure at the origin. Use variables to name any nonzero coordinates of vertices. Two possible placements of a triangle with a midsegment are shown below. In Figure 1, two of the vertices are on the x-axis with one of them at the origin. In Figure 2, two of the vertices are on the x-axis and the third vertex is on the y-axis. y y Q(a, b) M Q(0, 2b) N N M x P(0, 0) R(c, 0) Figure 1 x P( 2a, 0) O R(2c, 0) Figure 2 Multiplying the variable coordinates by 2, as in Figure 2, can make working with the Midpoint Formula easier. 302 Lesson 7-2 Applying Coordinate Geometry
Problem 1 Naming Coordinates How do you start the problem? Look at the position of the figure. Use the given information to determine how far each vertex is from the x- and y-axes. What are the coordinates of the vertices of each figure? A SQRE is a square where SQ 2a. B TRI is an isosceles triangle where TI 2a. The axes bisect each side. The y-axis is a median. y y S R Q x O E x T R O I Since SQRE is a square centered at The y-axis is a median, so it bisects TI . the origin and SQ 2a, S and Q are TI 2a, so T and I are both a units from the each a units from each axis. The y-axis. The height of TRI does not depend hsm11gmse 0608 t06543 hsm11gmse 0608 t06542 same is true for the other vertices. on a, so use a different variable for R. y y S( a, a) Q(a, a) R(0, b) x O E( a, a) x T( a, 0) O R(a, a) I(a, 0) Problem 2 TEKS Process Standard (1)(F) Using Variable Coordinates hsm11gmse 0608 t06544 hsm11gmse 0608 t06545 y The diagram shows a general parallelogram with a vertex at the origin and one side along the x-axis. What are the coordinates of D, the point of intersection of the diagonals of ABCO? How do you know? C(2b, 2c) B(2a 2b, 2c) D x The coordinates of the vertices of ABCO OB bisects AC, and AC bisects OB. O A(2a, 0) Since the diagonals of a parallelogram bisect each other, the midpoint of each segment is their point of intersection. Use the Midpoint Formula to find the midpoint of onehsm11gmse 0608 t06548 diagonal. The coordinates of D Use the Midpoint Formula to find the midpoint of AC. D midpoint of AC ( ) 2a 2b 0 2c (a b, c) 2 , 2 The coordinates of the point of intersection of the diagonals of ABCO are (a b, c). PearsonTEXAS.com 303
Problem 3 TEKS Process Standard (1)(D) Planning a Coordinate Proof How do you start? Start by drawing a diagram. Think about how you want to place the figure in the coordinate plane. Plan a coordinate proof of the Triangle Midsegment Theorem (Theorem 5-1), which states if a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half as long. Step 1 Draw and label a figure. Step 2 Write the Given and Prove statements. Midpoints will be involved, so use multiples of 2 to name coordinates. Use the information on the diagram to write the statements. y Given: M is the midpoint of PQ. N is the midpoint of QR. Q(0, 2b) Prove: M N MN } PR, MN 12PR x P( 2a, 0) O R(2c, 0) Step 3 Determine the formulas you will need. Then write the plan. First, use the Midpoint Formula to find the coordinates of M and N. T hen use the Slope Formula to determine whether the slopes of MN and PR are equal. If they are, MN and PR are parallel. NLINE HO ME RK O F inally, use the Distance Formula to find and compare the lengths of MN and PR. WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. What are the coordinates of the vertices of each figure? For additional support when completing your homework, go to PearsonTEXAS.com. 1. rectangle with base b 2. square with sides of 3. square centered at the origin, and height h length a with side length b y S S T y W x x O y T T O x W O Z S W 4. parallelogram where S is 5. rhombus centered at the 6. isosceles trapezoid with bases a units from the origin and origin, with SW 2r and centered at the origin, with Z is b units from the origin TZ 2t longer base 2a and OR c hsm11gmse 0608 t06552.ai hsm11gmse 0608 t06553.ai y y hsm11gmse 0608 t06551.ai y S T T x O 304 Z W Lesson 7-2 Applying Coordinate Geometry hsm11gmse 0608 t06554.ai O Z R T W x S W x S O Z hsm11gmse 0608 t06555.ai hsm11gmse 0608 t06556.ai
7. The diagram at the right shows a parallelogram. Without using the Distance Formula, determine whether the parallelogram is a rhombus. How do you know? y A( a, a) B(b, b) x O 8. Create Representations to Communicate Mathematical Ideas (1)(E) Place a general quadrilateral in the coordinate plane. D( b, b) 9. Analyze Mathematical Relationships (1)(F) A rectangle LMNP is centered at the origin with M(r, -s). What are the coordinates of P? 10. Plan a coordinate proof to show that the midpoints of the sides of an isosceles trapezoid form a rhombus. C(a, a) y hsm11gmse 0608 t06557.ai E a. Name the coordinates of isosceles trapezoid TRAP at the right, with bottom base length 4a, top base length 4b, and EG 2c. The y-axis bisects the bases. R A F D b. Write the Given and Prove statements. x T c. How will you find the coordinates of the midpoints of each side? P G d. How will you determine whether DEFG is a rhombus? 11. Analyze Mathematical Relationships (1)(F) Make two drawings of an isosceles triangle with base length 2b and height 2c. hsm11gmse 0608 t06558.ai a. In one drawing, place the base on the x-axis with a vertex at the origin. b. In the second, place the base on the x-axis with its midpoint at the origin. c. Find the lengths of the legs of the triangle as placed in part (a). d. Find the lengths of the legs of the triangle as placed in part (b). e. How do the results of parts (c) and (d) compare? 12. W and Z are the midpoints of OR and ST , respectively. In parts (a)–(c), find the coordinates of W and Z. y a. R(a, b) W O (?, ?) S(c, d) b. y R(2a, 2b) Z W x T(e, 0) O (?, ?) S(2c, 2d) c. y R(4a, 4b) Z W x T(2e, 0) (?, ?) O S(4c, 4d) Z x T(4e, 0) d. You are asked to plan a coordinate proof involving the midpoint of WZ. Which of figures (a)–(c) would you prefer to use? Explain. hsm11gmse 0608 t06562.ai hsm11gmse 0608 t06563.ai hsm11gmse 0608 t06564.ai 13. What property of a rhombus makes it convenient to place its diagonals on the x‑ and y‑axes? PearsonTEXAS.com 305
Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Plan the coordinate proof of each statement. 14. The opposite sides of a parallelogram are congruent (Theorem 6-3). 15. The diagonals of a rectangle bisect each other. 16. The consecutive sides of a square are perpendicular. Classify each quadrilateral as precisely as possible. 17. A(b, 2c), B(4b, 3c), C(5b, c), D(2b, 0) 18. O(0, 0), P(t, 2s), Q(3t, 2s), R(4t, 0) 19. E(a, b), F(2a, 2b), G(3a, b), H(2a, -b) 20. O(0, 0), L( -e, f), M(f - e, f e), N(f, e) STEM 21. Apply Mathematics (1)(A) Marine biologists sometimes use a coordinate system on the ocean floor. They record the coordinates of points where specimens are found. Assume that each diver searches a square area and can go no farther than b units from the starting point. Draw a model for the region one diver can search. Assign coordinates to the vertices without using any new variables. Here are coordinates for eight points in the coordinate plane (q p 0). A(0, 0), B(p, 0), C(q, 0), D(p q, 0), E(0, q), F(p, q), G(q, q), H(p q, q). Which four points, if any, are the vertices for each type of figure? 22. parallelogram 23. rhombus 24. rectangle TEXAS Test Practice 25. Which number of right angles is NOT possible for a quadrilateral to have? A. exactly one B. exactly two C. exactly three D. exactly four 26. The vertices of a rhombus are located at (a, 0), (0, b), ( -a, 0), and (0, -b), where a 7 0 and b 7 0. What is the midpoint of the side that is in Quadrant II? (a b) F. 2 , 2 ( a b G. - 2 , 2 ) ( a b H. - 2 , - 2 ) (a b J. 2 , - 2 ) 27. In PQRS, PQ 35 cm and QR 12 cm. What is the perimeter of PQRS? A. 23 cm B. 47 cm C. 94 cm D. 420 cm 28. In PQR, PQ 7 PR 7 QR. One angle measures 170. List all possible whole number values for m P. 306 Lesson 7-2 Applying Coordinate Geometry
Technology Lab Use With Lesson 7-3 Quadrilaterals in Quadrilaterals teks (5)(A), (1)(G) Construct Use geometry software to construct a quadrilateral ABCD. Construct the midpoint of each side of ABCD. Construct segments joining the midpoints, in order, to form quadrilateral EFGH. Investigate Measure the lengths of the sides of EFGH and their slopes. Measure the angles of EFGH. D G C H A F E B What kind of quadrilateral does EFGH appear to be? Exercises 1. Manipulate quadrilateral ABCD. a. Make a conjecture about the quadrilateral with vertices that are the midpoints of the sides of a quadrilateral. b. Does your conjecture hold when ABCD is concave? c. Can you manipulate ABCD so that your conjecture doesn’t hold? 2. Extend Draw the diagonals of ABCD. a. Describe EFGH when the diagonals are perpendicular. b. Describe EFGH when the diagonals are congruent. c. Describe EFGH when the diagonals are both perpendicular and congruent. 3. Construct the midpoints of EFGH and use them to construct quadrilateral IJKL. Construct the midpoints of IJKL and use them to construct quadrilateral MNOP. For MNOP and EFGH, compare the ratios of the lengths of the sides, perimeters, and areas. How are the sides of MNOP and EFGH related? 4. Writing In Exercise 1, you made a conjecture as to the type of quadrilateral EFGH appears to be. Prove your conjecture. Include in your proof the Triangle Midsegment Theorem, “If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and half its length.” 5. Describe the quadrilateral formed by joining the midpoints, in order, of the sides of each of the following. Justify each response. a. parallelogram b. rectangle c. rhombus d. square e. trapezoid f. isosceles trapezoid g. kite hsm11gmse 0609a t06183.ai D G K H C N J O L M P F I A E B hsm11gmse 0609a t06187.ai PearsonTEXAS.com 307
7-3 Proofs Using Coordinate Geometry TEKS FOCUS VOCABULARY TEKS (2)(B) Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines. TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated Representation – a way to display or Additional TEKS (1)(G), (6)(D) describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING You can prove geometric relationships using variable coordinates for figures in the coordinate plane. Problem 1 Proof TEKS Process Standard (1)(D) Proving Congruence of Medians Use coordinate geometry to prove that the medians drawn to the congruent sides of an isosceles triangle are congruent. y Given: PQR with PQ RQ, M is the midpoint of PQ, How can you verify congruency of the medians? You need to find the coordinates of the midpoints using the Midpoint Formula. Then you can find the lengths of the medians by using the Distance Formula. M Prove: PN RM Use the Midpoint Formula to find the coordinates of M and N. ( N ( M ) - 2a 0, 0 2b ( -a, b) 2 2 N x P( 2a, 0) O R(2a, 0) ) 2a 0, 0 2b (a, b) 2 2 Use the Distance Formula to find PN and RM. PN 2(a - ( -2a))2 (b - 0)2 29a2 b2 RM 2( -a - 2a)2 (b - 0)2 29a2 b2 Since PN RM, the two medians are congruent. 308 Q(0, 2b) N is the midpoint of RQ Lesson 7-3 Proofs Using Coordinate Geometry hsm11gmse 0609 t06604
Problem 2 Proof Refer to the plan from Lesson 7-2. Find the coordinates of M and N. Determine whether MN is parallel to PR. Then find and compare the lengths of MN and PR. Proving the Triangle Midsegment Theorem y Write a coordinate proof of the Triangle Midsegment Theorem (Theorem 5-1). Q(0, 2b) Given: M is the midpoint of PQ. M N N is the midpoint of QR. Prove: MN } PR, MN x P( 2a, 0) O 1 2 PR Statements Reasons 1) Given 1) M is the midpoint of PQ and N is the midpoint of QR. ( N ( 2) M R(2c, 0) ) - 2a 0, 0 2b ( -a, b) 2 2 ) 2) Midpoint Formula 0 2c, 2b 0 (c, b) 2 2 b-b 0 c - ( - a) 0-0 slope of PR 0 2c - ( - 2a) 3) slope of MN 3) Slope Formula 4) MN } PR 4) } lines have same slopes. 5) MN 2(c - ( -a))2 (b - b)2 c a 5) Distance Formula 6) PR 2MN 6) Substitution Property 7) MN 12 PR 7) Division Property of Equality PR 2(2c - ( -2a))2 (0 - 0)2 2(c a) NLINE HO ME RK O So MN } PR and MN 12 PR. WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Tell whether you can reach each type of conclusion below using coordinate methods for any points A, B, C, D, E, and F. Give a reason for each answer. For additional support when completing your homework, go to PearsonTEXAS.com. 1. AB CD 2. AB } CD 3. AB # CD 4. AB bisects CD. 5. AB bisects CAD. 6. A B 7. A is a right angle. 8. AB BC AC 9. Quadrilateral ABCD is a rhombus. 11. A is the supplement of B. 10. AB and CD bisect each other. 12. AB, CD, and EF are concurrent. PearsonTEXAS.com 309
Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Use coordinate geometry to prove each statement. y F( b, c) G(b, c) 13. The diagonals of an isosceles trapezoid are congruent. x Proof Given: Trapezoid EFGH with EF GH Prove: EG FH 14. The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. Proof Given: OEF is a right triangle, M is the midpoint of EF . O E( a, 0) H(a, 0) y E(0, 2b) hsm11gmse 0609 t06603 M Prove: EM FM OM x O 15. If two medians of a triangle are congruent, then the triangle is isosceles. Proof Given: PQR, M is the midpoint of PQ, N is the midpoint of RQ, PN RM Prove: PQ RQ 16. JKL has vertices J(2, 4), K(2, –4), and L(–6, 4), with JK JL. Point M is the midpoint of JK , and point N is the midpoint of JL. Verify that KN LM. F(2a, 0) y Q(0, 2b) hsm11gmse 0609 t06327 M N x P( 2a, 0) O R(2c, 0) Use coordinate geometry to prove each statement. Proof 17. If a parallelogram is a rhombus, its diagonals are perpendicular (Theorem 6-13). hsm11gmse 0609 t06604 18. The altitude to the base of an isosceles triangle bisects the base. 19. If the midpoints of a trapezoid are joined to form a quadrilateral, then the quadrilateral is a parallelogram. 20. One diagonal of a kite divides the kite into two congruent triangles. 21. Apply Mathematics (1)(A) The flag design at the right is made by connecting the midpoints of the sides of a rectangle. Use coordinate geometry to prove that the quadrilateral formed is a rhombus. Proof 22. Connect Mathematical Ideas (1)(F) Give an example of a statement that you think is easier to prove with a coordinate geometry proof than with a proof method that does not require coordinate geometry. Explain your choice. 310 Lesson 7-3 Proofs Using Coordinate Geometry
23. Complete the steps to prove Theorem 5-8 which states that the centroid of a triangle is two thirds t
coordinate proof, p. 302 prueba de coordenadas VOCABULARY 7-1Polygons in the Coordinate Plane 7-2 Applying Coordinate Geometry 7-3Proofs Using Coordinate Geometry TOPIC OVERVIEW 294 Topic 7 Coordinate Geometry. 3-Act Math 3-Act Math If You Need Help . . . Vocabulary online
Placing Figures in a Coordinate Plane A coordinate proof involves placing geometric fi gures in a coordinate plane. When you use variables to represent the coordinates of a fi gure in a coordinate proof, the results are true for all fi gures of that type. Placing a Figure in a Coordinate Plane Place each fi gure in a coordinate plane in a way .
Placing Figures in a Coordinate Plane A coordinate proof involves placing geometric fi gures in a coordinate plane. When you use variables to represent the coordinates of a fi gure in a coordinate proof, the results are true for all fi gures of that type. Placing a Figure in a Coordinate Plane Place each fi gure in a coordinate plane in a way .
Topic 5: Not essential to progress to next grade, rather to be integrated with topic 2 and 3. Gr.7 Term 3 37 Topic 1 Dramatic Skills Development Topic 2 Drama Elements in Playmaking Topic 1: Reduced vocal and physical exercises. Topic 2: No reductions. Topic 5: Topic 5:Removed and integrated with topic 2 and 3.
Timeframe Unit Instructional Topics 4 Weeks Les vacances Topic 1: Transportation . 3 Weeks Les contes Topic 1: Grammar Topic 2: Fairy Tales Topic 3: Fables Topic 4: Legends 3 Weeks La nature Topic 1: Animals Topic 2: Climate and Geography Topic 3: Environment 4.5 Weeks L’histoire Topic 1: Pre-History - 1453 . Plan real or imaginary travel .
PROVING RECTANGLES USING COORDINATE GEOMETRY WAYS TO PROVE HOW ARE WE GOING TO DO THIS USING COORDINATE GEOMETRY? 1.) Prove that it is a PARALLELOGRAM with 2.) Prove that it is a PARALLELOGRM with 3.) Prove that all angles are Let’s pick a way and stick with it! - - 2.) Find the coordinates
Four coordinate systems in navigation system are considered: global coordinate system, tractor coordinate system, IMU coordinate system and GPS-receiver coordinate system, as superscripts or subscripts YKJ, t, i and GPS. Tractor coordinate system origin is located at the ground level of the tractor, under the center point of the
EXAMPLE 1 coordinate proof GOAL 1 Place geometric figures in a coordinate plane. Write a coordinate proof. Sometimes a coordinate proof is the most efficient way to prove a statement. Why you should learn it GOAL 2 GOAL 1 What you should learn 4.7 Placing Figures in a Coordinate Plane Draw a right triangle with legs of 3 units and 4 units on a .
Business Architecture. o Descriptions of an Organization (Business model, MSGs, operating model). o Business processes & workflows. o Stakeholders and their roles and relationships. o Business rules (what the actors must do in a BP) o A lot about Process Change o Process Reengineering & Process Change, o Quality Movement o BPMN, UML Use Case Models . Lecture 4: Business Process Redesign .
