Coordinate Geometry VOCABULARY. English/Spanish Vocabulary Audio Online:

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1 Topic 7 Coordinate Geometr TPIC VERVIEW VCABULARY 7-1 Polgons in the Coordinate Plane 7- Appling Coordinate Geometr 7-3 Proofs Using Coordinate Geometr English/panish Vocabular Audio nline: English panish coordinate geometr, p. 96 geometría de coordenadas coordinate proof, p. 30 prueba de coordenadas DIGITAL APP PRINT and ebook Access Your Homework... nline homework You can do all of our homework online with built-in eamples and how Me How support! When ou log in to our account, ou ll see the homework our teacher has assigned ou. Your Digital Resources PearsonTEXA.com Homework Tutor app Do our homework anwhere! You can access the Practice and Application Eercises, as well as Virtual Nerd tutorials, with this Homework Tutor app, available on an mobile device. TUDENT TEXT AND Homework Helper Access the Practice and Application Eercises that ou are assigned for homework in the tudent Tet and Homework Helper, which is also available as an electronic book. 94 Topic 7 Coordinate Geometr

2 You Be the Judge 3-Act Math 3-Act Math Have ou ever been a judge in a contest or competition? What criteria did ou use to decide the winner? If ou were one of man judges, did ou all agree on who should win? ften 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 the are rating. Think about this as ou watch the 3-Act Math video. can page to see a video for this 3-Act Math Task. If You Need Help... Vocabular nline You ll find definitions of math terms in both English and panish. All of the terms have audio support. Interactive eploration You ll have access to a robust assortment of interactive eplorations, including interactive concept eplorations, dnamic activitites, and topiclevel eploration activities. Learning Animations You can also access all of the stepped-out learning animations that ou studied in class. tudent Companion Refer to our notes and solutions in our tudent Companion. Remember that our tudent Companion is also available as an ACTIVebook accessible on an digital device. Interactive Math tools These interactive math tools give ou opportunities to eplore in greater depth ke concepts to help build understanding. Virtual Nerd Not sure how to do some of the practice eercises? Check out the Virtual Nerd videos for stepped-out, multi-level instructional support. PearsonTEXA.com 95

3 TEK FCU 7-1 Polgons in the Coordinate Plane VCABULARY TEK ()(B) Derive and use the distance, slope, and midpoint formulas to verif geometric relationships, including congruence of segments and parallelism or perpendicularit of pairs of lines. TEK (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, diagrams, graphs, and language as appropriate. Additional TEK (1)(F), (1)(G), (6)(E) Coordinate geometr the analtical use of algebra to stud geometric properties of figures drawn on the coordinate plane Implication a conclusion that follows from previousl stated ideas or reasoning without being eplicitl stated Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. EENTIAL UNDERTANDING You can classif figures in the coordinate plane using the formulas for slope, distance, and midpoint. Ke Concept Formulas and the Coordinate Plane Formula Distance Formula d = ( - 1 ) + ( - 1 ) When to Use It To determine whether sides are congruent diagonals are congruent Midpoint Formula M = ( 1 +, 1 + ) To determine the coordinates of the midpoint of a side whether diagonals bisect each other lope Formula m = To determine whether opposite sides are parallel diagonals are perpendicular sides are perpendicular 96 Lesson 7-1 Polgons in the Coordinate Plane

4 Problem 1 TEK Process tandard (1)(D) How do ou classif 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 an sides are congruent. Classifing a Triangle Is ABC scalene, isosceles, or equilateral? The vertices of the triangle are A(0, 1), B(4, 4), and C(7, 0). Use the Distance Formula to find the lengths of the sides. AB = (4-0) + (4-1) implif within parentheses. = Then simplif the powers. = 15 = 5 implif. 4 A B 4 6 C BC = (7-4) + (0-4) implif within parentheses. = Then simplif the powers. = 15 = 5 implif. CA = (0-7) + (1-0) implif within parentheses. = Then simplif the powers. = 150 = 51 implif. ince AB = BC = 5, two sides of the triangle are congruent. B definition, ABC is isosceles. Problem TEK Process tandard (1)(G) Classifing a Quadrilateral How can ou show that a quadrilateral is a rhombus? First, prove the quadrilateral is a parallelogram. Net, show that its diagonals are perpendicular. Prove ABCD is a rhombus. tep 1 Use the lope Formula to verif the opposite sides are parallel. 4-0 slope of AB = (-) = 4 slope of CD = = 4 slope of BC = (-1) = 1 4 slope of AD = (-) = 1 4 AB } CD and BC } AD, so ABCD is a parallelogram. A B 6 C D 4 tep Use the lope Formula to verif the diagonals are perpendicular. slope of AC = = 1 slope of BD = 3 - (-) - (-1) = -1 The product of the slopes of the diagonals is -1, so AC # BD. ince quadrilateral ABCD is a parallelogram with perpendicular diagonals, it is a rhombus (Theorem 6-16). PearsonTEXA.com 97

5 Problem 3 Besides using the lope Formula to verif that opposite sides are parallel, how can ou show that NPQR is a parallelogram? You can show it has two pairs of congruent opposite sides, or that its diagonals bisect each other. Verifing Parallelism of Line egments Without using the lope Formula, verif that NP } QR and PQ } NR. Method 1 N Use the Distance Formula. P R 4 - First, find the distances between points N and P and between points R and Q to show that NP and QR are congruent. Q Then find the distances between points P and Q and between points R and N to show that PQ and NR are congruent. NP = (- - (-5)) + (3 - ) PQ = (1 - (-)) + (0-3) = = = 10 = 18, or 3 QR = (1 - (-)) + (0 - (-1)) NR = (-5 - (-)) + ( - (-1)) = = = 10 = 18, or 3 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. B the definition of a parallelogram, NP } QR and PQ } NR. Method Use the Midpoint Formula. Find the midpoints of NQ and PR to determine whether the are the same point. midpoint of NQ = ( , + 0 ) = (-, 1) midpoint of PR = ( - + (-), 3 + (-1) ) = (-, 1) 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. B the definition of a parallelogram, NP } QR and PQ } NR. 98 Lesson 7-1 Polgons in the Coordinate Plane

6 Problem 4 Verifing Congruence of egments How can ou approach Problem 4 without using the Distance Formula? Find another wa to verif that segments are congruent. Use the Perpendicular Bisector Theorem (Theorem 5-). M is the midpoint of DE. Without using the Distance Formula, verif that FD FE. tep 1 tep Use the Midpoint Formula to find the coordinates of M. M = ( (-1), ) = ( - 7, 7 ) Use the lope Formula to show FM # DE. 7 slope of FM = (-) = - 3 = -1 E 6 M 4 D F slope of DE = (-6) = 5 5 = 1 The product of the slopes is -1, so FM # DE. M is the midpoint of DE, so FM is the perpendicular bisector of DE. The Perpendicular Bisector Theorem (Theorem 5-) states if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. ince point F is on the perpendicular bisector of DE, then F is equidistant from the endpoints, D and E. Therefore, FD FE. NLINE H M E W R K PRACTICE and APPLICATIN EXERCIE can page for a Virtual Nerd tutorial video. For additional support when completing our homework, go to PearsonTEXA.com. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Determine whether ABC is scalene, isosceles, or equilateral. Eplain. 1. A. A 3. C B C B A C B 4. Eplain Mathematical Ideas (1)(G) PQR has vertices P( -4, 4), Q(, 0), R(0, -3), ( -6, 1). Without using the lope Formula, verif PQ } R and QR } P. Use the Distance Formula in our solution. 5. Eplain Mathematical Ideas (1)(G) PQR has vertices P(8, 5), Q(5, -4), R( -1, -), (, 7). Without using the lope Formula, verif PQ } R and QR } P. Use the Midpoint Formula in our solution. 6. Eplain Mathematical Ideas (1)(G) An isosceles triangle has vertices A(3, 3), B(8, 4), C(, -). M is the midpoint of BC. Without using the Distance Formula, verif that AB AC. Use the Perpendicular Bisector Theorem (Theorem 5-) in our solution. PearsonTEXA.com 99

7 Determine whether the parallelogram is a rhombus, rectangle, square, or none of these. Eplain. 7. P(-1, ), (0, 0), (4, 0), T(3, ) 8. L(1, ), M(3, 3), N(5, ), P(3, 1) 9. R(-, -3), (4, 0), T(3, ), V(-3, -1) 10. W(-3, 0), I(0, 3), N(3, 0), D(0, -3) 11. Appl Mathematics (1)(A) An artist is planning to paint a rectangle on a wall as part of a mural. Quadrilateral PQR in the coordinate grid at the right represents the planned location of the rectangle. Is PQR a rectangle? If so, eplain our reasoning. If not, describe how the artist could change the plans to make sure PQR is a rectangle. 1. Justif Mathematical Arguments (1)(G) A classmate sas that if ou can show that quadrilateral EFGH is a rhombus, then ou onl need to show that one pair of adjacent sides is perpendicular in order to prove that EFGH is a square. Is the classmate correct? Eplain our reasoning. 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. R P Q 13. T(1, 1), R(3, 8), I(6, 4) 14. J(-5, 0), K(5, 8), L(4, -1) 15. A(3, ), B(-10, 4), C(-5, -8) 16. H(1, -), 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 ou know? 18. Displa 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? Eplain. Q P T R 6 W Graph and label each quadrilateral with the given vertices. Then determine the most precise name for each quadrilateral. 19. P(-5, 0), Q(-3, ), R(3, ), (5, 0) 0. (0, 0), T(4, 0), U(3, ), V(-1, ) 1. F(0, 0), G(5, 5), H(8, 4), I(7, 1). M(-14, 4), N(1, 6), P(3, -9), Q(-1, -11) 3. A(3, 5), B(7, 6), C(6, ), D(, 1) 4. N(-6, 4), P(-3, 1), Q(0, ), R(-3, 5) 5. J(, 1), K(5, 4), L(8, 1), M(, -3) 6. H(-, -3), I(4, 0), J(3, ), K(-3, -1) 7. DE is a midsegment of ABC at the right. how that the Triangle Midsegment Theorem (Theorem 5-1) holds true for ABC. 8. a. Describe two was ou can show whether a quadrilateral in the coordinate plane is a square. b. Evaluate Reasonableness (1)(B) Which method is more efficient? Eplain. A B D C E Lesson 7-1 Polgons in the Coordinate Plane

8 9. Appl 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. he wants the chairs oriented at the vertices of a parallelogram. Does she need to fi 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(, 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 n G 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 F b. What is the most precise classification of EFGH? c. Are ABCD and EFGH congruent? Eplain. 4 6 B C 4 E H hsm11gmse_0607_t0659 TEXA Test Practice 36. In the diagram, lines / and m are parallel. What is the value of? A. 5 C. 13 B. 1 D m ( 11) 37. K( -3, 0), I(0, ), and T(3, 0) are three vertices of a kite. Which point could be the fourth verte? F. E(0, 5) G. E(0, 0) H. E(0, -) hsm11gmse_0607_t06593 J. E(0, -10) 38. In the diagram, which segment is shortest? A. P C. PQ B. PR D. QR 39. A( -3, 1), B( -1, -), and C(, 1) are three vertices of quadrilateral ABCD. Could ABCD be a rectangle? Eplain. P 57 Q R PearsonTEXA.com hsm11gmse_0607_t

9 TEK FCU 7- Appling Coordinate Geometr VCABULARY TEK Foundational to () The student uses the process skills to understand the connections between algebra and geometr and uses the one- and twodimensional coordinate sstems to verif geometric conjectures. TEK (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, 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 previousl stated ideas or reasoning without being eplicitl stated Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. Additional TEK (1)(F) EENTIAL UNDERTANDING You can use variables to name the coordinates of a figure. This allows ou to show that relationships are true for a general case. Ke Concept Using Variables for Coordinates To place a general figure in the coordinate plane, it is usuall helpful to place one side on an ais or to center the figure at the origin. Use variables to name an 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 -ais with one of them at the origin. In Figure, two of the vertices are on the -ais and the third verte is on the -ais. Q(a, b) M N P(0, 0) R(c, 0) Figure 1 Q(0, b) M N P( a, 0) R(c, 0) Figure Multipling the variable coordinates b, as in Figure, can make working with the Midpoint Formula easier. 30 Lesson 7- Appling Coordinate Geometr

10 Problem 1 Naming Coordinates How do ou start the problem? Look at the position of the figure. Use the given information to determine how far each verte is from the - and -aes. What are the coordinates of the vertices of each figure? A QRE is a square where Q = a. B TRI is an isosceles triangle where TI = a. The aes bisect each side. The -ais is a median. E Q R T R I ince QRE is a square centered at The -ais is a median, so it bisects TI. the origin and Q = a, and Q are TI = a, so T and I are both a units from the each a units from each ais. The -ais. The height of TRI does not depend same is true for the other vertices. on a, so use a different variable for R. ( a, a) Q(a, a) R(0, b) E( a, a) R(a, a) T( a, 0) I(a, 0) Problem TEK Process tandard (1)(F) Using Variable Coordinates The diagram shows a general parallelogram with a verte at the origin and one side along the -ais. What are the coordinates of D, the point of intersection of the diagonals of ABC? How do ou know? C(b, c) D B(a b, c) The coordinates of the vertices of ABC B bisects AC, and AC bisects B. A(a, 0) The coordinates of D ince 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 one diagonal. Use the Midpoint Formula to find the midpoint of AC. D = midpoint of AC = ( a + b, 0 + c ) = (a + b, c) The coordinates of the point of intersection of the diagonals of ABC are (a + b, c). PearsonTEXA.com 303

11 Problem 3 TEK Process tandard (1)(D) How do ou start? tart b drawing a diagram. Think about how ou want to place the figure in the coordinate plane. Planning a Coordinate Proof 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. tep 1 Draw and label a figure. tep Write the Given and Prove statements. Midpoints will be involved, so use Use the information on the multiples of to name coordinates. diagram to write the statements. Q(0, b) Given: M is the midpoint of PQ. N is the midpoint of QR. Prove: M N MN } PR, MN = 1 PR P( a, 0) R(c, 0) tep 3 Determine the formulas ou will need. Then write the plan. First, use the Midpoint Formula to find the coordinates of M and N. Then use the lope Formula to determine whether the slopes of MN and PR are equal. If the are, MN and PR are parallel. Finall, use the Distance Formula to find and compare the lengths of MN and PR. NLINE H M E W R K PRACTICE and APPLICATIN EXERCIE can page for a Virtual Nerd tutorial video. For additional support when completing our homework, go to PearsonTEXA.com. What are the coordinates of the vertices of each figure? 1. rectangle with base b. square with sides of 3. square centered at the origin, and height h length a with side length b T T W T W W Z 4. parallelogram where is 5. rhombus centered at the 6. isosceles trapezoid with bases a units from the origin and origin, with W = r and centered at the origin, with Z is b units from the origin TZ = t longer base a and R = c T T T R W W Z W Z Z 304 Lesson 7- Appling Coordinate Geometr

12 7. The diagram at the right shows a parallelogram. Without using the Distance Formula, determine whether the parallelogram is a rhombus. How do ou know? 8. Create Representations to Communicate Mathematical Ideas (1)(E) Place a general quadrilateral in the coordinate plane. 9. Analze 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. a. Name the coordinates of isosceles trapezoid TRAP at the right, with bottom base length 4a, top base length 4b, and EG = c. The -ais bisects the bases. D b. Write the Given and Prove statements. c. How will ou find the coordinates of the midpoints of each side? d. How will ou determine whether DEFG is a rhombus? A( a, a) D( b, b) 11. Analze Mathematical Relationships (1)(F) Make two drawings of an isosceles triangle with base length b and height c. a. In one drawing, place the base on the -ais with a verte at the origin. b. In the second, place the base on the -ais 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? 1. W and Z are the midpoints of R and T, respectivel. In parts (a) (c), find the coordinates of W and Z. a. b. c. R(a, b) (c, d) R(a, b) (c, d) R(4a, 4b) R E B(b, b) A C(a, a) T G P F (4c, 4d) W (?,?) Z T(e, 0) W (?,?) Z T(e, 0) W (?,?) Z T(4e, 0) d. You are asked to plan a coordinate proof involving the midpoint of WZ. Which of figures (a) (c) would ou prefer to use? Eplain. 13. What propert of a rhombus makes it convenient to place its diagonals on the and aes? PearsonTEXA.com 305

13 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. Classif each quadrilateral as precisel as possible. 17. A(b, c), B(4b, 3c), C(5b, c), D(b, 0) 18. (0, 0), P(t, s), Q(3t, s), R(4t, 0) 19. E(a, b), F(a, b), G(3a, b), H(a, -b) 0. (0, 0), L( -e, f), M(f - e, f + e), N(f, e) TEM 1. Appl Mathematics (1)(A) Marine biologists sometimes use a coordinate sstem on the ocean floor. The 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 an 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 an, are the vertices for each tpe of figure?. parallelogram 3. rhombus 4. rectangle TEXA Test Practice 5. Which number of right angles is NT possible for a quadrilateral to have? A. eactl one B. eactl two C. eactl three D. eactl four 6. 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? F. ( a, b ) G. ( - a, b ) H. ( - a, - b ) J. ( a, - b ) 7. In PQR, PQ = 35 cm and QR = 1 cm. What is the perimeter of PQR? A. 3 cm B. 47 cm C. 94 cm D. 40 cm 8. In PQR, PQ 7 PR 7 QR. ne angle measures 170. List all possible whole number values for m P. 306 Lesson 7- Appling Coordinate Geometr

14 Technolog Lab Quadrilaterals in Quadrilaterals Use With Lesson 7-3 teks (5)(A), (1)(G) Construct Use geometr 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. What kind of quadrilateral does EFGH appear to be? A H D E G C F B Eercises 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 our conjecture hold when ABCD is concave? c. Can ou manipulate ABCD so that our conjecture doesn t hold?. Etend 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 MNP. For MNP and EFGH, compare the ratios of the lengths of the sides, perimeters, and areas. How are the sides of MNP and EFGH related? D H A L K G N P J M I F C 4. Writing In Eercise 1, ou made a conjecture as to the tpe of quadrilateral EFGH appears to be. Prove our conjecture. Include in our 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. E B 5. Describe the quadrilateral formed b joining the midpoints, in order, of the sides of each of the following. Justif each response. a. parallelogram b. rectangle c. rhombus d. square e. trapezoid f. isosceles trapezoid g. kite PearsonTEXA.com 307

15 7-3 Proofs Using Coordinate Geometr TEK FCU VCABULARY TEK ()(B) Derive and use the distance, slope, and midpoint formulas to verif geometric relationships, including congruence of segments and parallelism or perpendicularit of pairs of lines. TEK (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, diagrams, graphs, and language as appropriate. Implication a conclusion that follows from previousl stated ideas or reasoning without being eplicitl stated Representation a wa to displa or Additional TEK (1)(G), (6)(D) describe information. You can use a representation to present mathematical ideas and data. EENTIAL UNDERTANDING You can prove geometric relationships using variable coordinates for figures in the coordinate plane. Problem 1 Proof TEK Process tandard (1)(D) Proving Congruence of Medians Use coordinate geometr to prove that the medians drawn to the congruent sides of an isosceles triangle are congruent. Given: PQR with PQ RQ, M is the midpoint of PQ, How can ou verif congruenc of the medians? You need to find the coordinates of the midpoints using the Midpoint Formula. Then ou can find the lengths of the medians b using the Distance Formula. M Prove: PN RM Use the Midpoint Formula to find the coordinates of M and N. ( N=( M= ) - a + 0, 0 + b = ( -a, b) N P( a, 0) R(a, 0) ) a + 0, 0 + b = (a, b) Use the Distance Formula to find PN and RM. PN = (a - ( -a)) + (b - 0) = 9a + b RM = ( -a - a) + (b - 0) = 9a + b ince PN = RM, the two medians are congruent. 308 Q(0, b) N is the midpoint of RQ Lesson 7-3 Proofs Using Coordinate Geometr hsm11gmse_0609_t06604

16 Problem Refer to the plan from Lesson 7-. Find the coordinates of M and N. Determine whether MN is parallel to PR. Then find and compare the lengths of MN and PR. Proof Proving the Triangle Midsegment Theorem Write a coordinate proof of the Triangle Midsegment Theorem (Theorem 5-1). Given: M is the midpoint of PQ. N is the midpoint of QR. Prove: MN } PR, MN = 1 PR tatements Reasons 1) M is the midpoint of PQ and 1) Given N is the midpoint of QR. ) M = ( - a + 0, ) = (-a, b) ) Midpoint Formula N = ( 0 + c, b + 0 ) = (c, b) 3) slope of MN = b - b c - (-a) = 0 3) lope Formula 0-0 slope of PR = c - (-a) = 0 4) MN } PR 4) } lines have same slopes. 5) MN = (c - (-a)) + (b - b) = c + a PR = (c - (-a)) + (0-0) = (c + a) 5) Distance Formula Q(0, b) M N P( a, 0) R(c, 0) 6) PR = MN 6) ubstitution Propert 7) MN = 1 PR 7) Division Propert of Equalit o MN } PR and MN = 1 PR. NLINE H M E W R K PRACTICE and APPLICATIN EXERCIE can page for a Virtual Nerd tutorial video. Tell whether ou can reach each tpe of conclusion below using coordinate methods for an points A, B, C, D, E, and F. Give a reason for each answer. For additional support when completing our homework, go to PearsonTEXA.com. 1. AB CD. 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. 10. AB and CD bisect each other. 11. A is the supplement of B. 1. AB, CD, and EF are concurrent. PearsonTEXA.com 309

17 Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Use coordinate geometr to prove each statement. F( b, c) G(b, c) 13. The diagonals of an isosceles trapezoid are congruent. Proof Given: Trapezoid EFGH with EF GH Prove: EG FH E( a, 0) H(a, 0) 14. The midpoint of the hpotenuse of a right triangle is Proof equidistant from the three vertices. E(0, b) Given: EF is a right triangle, M is the midpoint of EF. M Prove: EM = FM = M 15. If two medians of a triangle are congruent, then the triangle is Proof isosceles. Given: PQR, M is the midpoint of PQ, N is the midpoint of RQ, PN RM Prove: PQ RQ M F(a, 0) Q(0, b) N 16. JKL has vertices J(, 4), K(, 4), and L( 6, 4), with JK JL. Point M is the midpoint of JK, and point N is the midpoint of JL. Verif that KN LM. P( a, 0) R(c, 0) Use coordinate geometr to prove each statement. Proof 17. If a parallelogram is a rhombus, its diagonals are perpendicular (Theorem 6-13). 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. 0. ne diagonal of a kite divides the kite into two congruent triangles. 1. Appl Mathematics (1)(A) The flag design at the right Proof is made b connecting the midpoints of the sides of a rectangle. Use coordinate geometr to prove that the quadrilateral formed is a rhombus.. Connect Mathematical Ideas (1)(F) Give an eample of a statement that ou think is easier to prove with a coordinate geometr proof than with a proof method that does not require coordinate geometr. Eplain our choice. 310 Lesson 7-3 Proofs Using Coordinate Geometr

18 3. Complete the steps to prove Theorem 5-8 which states that the centroid of a Proof triangle is two thirds the distance from each verte to the midpoint of the opposite side. a. Find the coordinates of points L, M, and N, the midpoints of the sides of ABC. B(6q, 6r) b. Find equations of < AM >, < BN >, and < CL >. c. Find the coordinates of point P, the intersection of < AM > and < BN >. d. how that point P is on < CL >. e. Use the Distance Formula to show that point P is two thirds the distance from each verte to the midpoint of the opposite side. 4. Complete the steps to prove Theorem 5-9. You are given ABC with altitudes Proof p, q, and r. how that p, q, and r intersect at a point (called the orthocenter). c a. The slope of BC is - b. What is the slope of line p? b. how that the equation of line p is = b c ( - a). C(0, c) r c. What is the equation of line q? d. how that lines p and q intersect at (0, - c ab ). A(a, 0) e. Repeat parts (a) (c) to show that lines r and q intersect at (0, - c ab q ). f. What are the coordinates of the orthocenter of ABC? 5. Prove: If two lines are perpendicular, the product of their slopes is -1. Proof a. Two nonvertical lines, / 1 and /, intersect as shown at the right. Find the coordinates of C. B(, ) b. Choose coordinates for D and B. (Hint: Find the relationship between 1,, and 3. Then use congruent triangles.) c. Complete the proof that the product of slopes is -1. A L D(, ) 3 P N M p A(a, b) C(6p, 0) B(b, 0) 1 1 C(, ) TEXA Test Practice 6. In the diagram of PR at the right, P = 16 and R = 1. What is M? 7. ne endpoint of a segment is (7, -3). The midpoint is (3, 4). What is the length of the segment? 8. FGHI has sides with lengths FG = + 5, GH = + 7, HI = 3 -, and FI =. What is the length of the longer sides of FGHI? 9. In ABC, m A = 55. If m C is twice m A, what is m B? 16 8 P M 8 R 16 PearsonTEXA.com 311

19 Topic 7 Review TPIC VCABULARY coordinate geometr, p. 96 coordinate proof, p. 30 Check Your Understanding Choose the vocabular term that correctl completes the sentence. 1. ometimes it is easier to show that a theorem is true b using a? rather than a standard deductive proof..? uses algebra to stud geometric properties in a coordinate sstem. 7-1 Polgons in the Coordinate Plane Quick Review To determine whether sides or diagonals are congruent, use the Distance Formula. To determine the coordinate of the midpoint of a side, or whether the diagonals bisect each other, use the Midpoint Formula. To determine whether opposite sides are parallel, or whether diagonals or sides are perpendicular, use the lope Formula. Eercises Determine whether ABC is scalene, isosceles, or equilateral. 3. ( 1, 1) Eample XYZ has vertices X(1, 0), Y(, 4), and Z(4, 4). Is XYZ scalene, isosceles, or equilateral? To find the lengths of the legs, use the Distance Formula. XY = (- - 1) + (-4-0) = = 5 YZ = (4 - (-)) + (-4 - (-4)) = = 6 4. ( 1, ) (3, ) (3, 3) (0, ) 4 (1, 1) XZ = (4-1) + (-4-0) = = 5 Two side lengths are equal, so XYZ is isosceles. What is the most precise classification of the quadrilateral? 5. G(, 5), R(5, 8), A( -, 1), D( -5, 9) 6. F( -13, 7), I(1, 1), N(15, 7), E(1, -5) 7. Q(4, 5), U(1, 14), A(0, 5), D(1, -4) 8. W( -11, 4), H( -9, 10), A(, 10), T(4, 4) 31 Topic 7 Review

20 7- and 7-3 Coordinate Geometr and Coordinate Proofs Quick Review When placing a figure in the coordinate plane, it is usuall helpful to place at least one side on an ais. Use variables when naming the coordinates of a figure in order to show that relationships are true for a general case. Eample Rectangle PQR has length a and width 4b. The -ais bisects P and QR. What are the coordinates of the vertices? P ince the width of PQR is 4b and the -ais bisects P and QR, all the vertices are b units from the -ais. P is on the -ais, so P = (0, b) and = (0, -b). The length of PQR is a, so Q = (a, b) and R = (a, -b). Q R Eercises 9. In rhombus FLP, the aes form the diagonals. If L = a and FP = 4b, what are the coordinates of the vertices? F L 10. The figure at the right is a parallelogram. Give the coordinates of point P without using an new variables. ( b, c) ( a, 0) 11. Use coordinate geometr to prove that the quadrilateral formed b connecting the midpoints of a kite is a rectangle. P P PearsonTEXA.com 313

21 Topic 7 TEK Cumulative Practice Multiple Choice Read each question. Then write the letter of the correct answer on our paper. 1. How would ou classif RT? A. right B. isosceles C. scalene D. equilateral. Which equation represents the line that contains the perpendicular bisector of the segment shown? F. = - 3 G = 3 H. = 3 J. + = 3 3. A line passes through (3, -4) and has a slope of -5. How can ou find the -intercept of the graph? A. ubstitute -5 for b, -4 for, and 3 for in = m + b. Then solve for m, the -intercept. B. ubstitute -5 for b, 3 for, and -4 for in = m + b. Then solve for m, the -intercept. C. ubstitute -5 for m, -4 for, and 3 for in = m + b. Then solve for b, the -intercept. D. ubstitute -5 for m, 3 for, and -4 for in = m + b. Then solve for b, the -intercept. R T (1, ) (5, ) 4. TUV is a parallelogram. What are the coordinates of point U? F. (, ) G. ( + z, ) T(0, ) U(?,?) H. (, z) J. (z, ) 5. The diagram shows a general parallelogram with a verte at the origin and one side along the -ais. What are the coordinates of the point of intersection of the diagonals of the parallelogram? D(b, c) A(0, 0) B(a, 0) A. (a + b, c) B. (c, a + b) C. (a + b, c) D. (a + b, c) C(a + b, c) 6. PQR has vertices P(0, 0), Q(4, ), and (4, -). Its diagonals intersect at H(4, 0). What are the possible coordinates of R for PQR to be a kite? F. (8, 0) H. (4, 0) G. (0, 4) J. (10, 0) (, 0) V( z, 0) 314 Topic 7 TEK Cumulative Practice

22 Gridded Response 7. What is the slope of a line perpendicular to = 15? 8. What is the slope of the line containing the points (-, 5) and (4, -4)? 9. Am is designing a ramp up to a 16-in.-high skateboarding platform, as shown on the figure below. If she wants the slope of the ramp to be 1 3, what value should she choose for the -coordinate at the top of the ramp? (, 16) (60, 16) 13. A(3, -), B(5, 4), C(3, 6), D(1, 4) 14. A(1, -4), B(1, 1), C( -, ), D(-, -3) For Eercises 15 and 16, give the coordinates for points and T without using an new variables. Then find the midpoint and the slope of T. 15. rectangle T (a, b) (a, b) (60, 0) 16. parallelogram (c, d) T 10. What is m R in parallelogram RTU? Constructed Response R Prove that GHIJ is a rhombus. G H Graph each quadrilateral ABCD. Then determine the most precise name for it. 1. A(1, ), B(11, ), C(7, 5), D(4, 5) J 6 T I U (b, 0) 17. ketch two noncongruent parallelograms ABCD and EFGH such that AC BD EG FH. 18. Prove that the diagonals of square ABCD are congruent. 19. Write the coordinates of four points that determine each figure with the given conditions. ne verte is at the origin and one side is 3 units long. a. square c. rectangle b. parallelogram d. trapezoid D(0, a) A B(a, 0) 0. A parallelogram has vertices L(-, 5), M(3, 3), and N(1, 0). What are possible coordinates for its fourth verte? Eplain. C(a, a) PearsonTEXA.com 315