2.4. Classifying Figures on a Coordinate Grid. LEARN ABOUT the Math. Connecting slopes and lengths of line segments to classifying a figure

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.4 Classifying Figures n a Crdinate Grid YOU WILL NEED grid paper and ruler, r dynamic gemetry sftware GOAL Use prperties f line segments t classify tw-dimensinal figures.

1 .4 Classifying Figures n a Crdinate Grid YOU WILL NEED grid paper and ruler, r dynamic gemetry sftware GOAL Use prperties f line segments t classify tw-dimensinal figures. LEARN ABOUT the Math A surveyr has marked the crners f a lt where a building is ging t be cnstructed. The crners have crdinates P(-5,-5), Q(-30, 10), R(-5, 5), and S(0, 10). Each unit represents 1 m. The builder wants t knw the perimeter and shape f this building lt.? Hw can the builder use the crdinates f the crners t determine the shape and perimeter f the lt? EXAMPLE 1 Cnnecting slpes and lengths f line segments t classifying a figure Use analytic gemetry t identify the shape f quadrilateral PQRS and its perimeter. Anita s Slutin analytic gemetry gemetry that uses the y-aes, algebra, and equatins t describe relatins and psitins f gemetric figures R 30 y I pltted the pints n grid paper and then jined them t draw the figure. 0 Q 10 S P -10 I saw that the shape f the building lt lked like a parallelgram r a rhmbus, but I culdn t be sure. I knew that if PQRS was either f these figures, the ppsite sides wuld be parallel. I als knew that if PQRS was a rhmbus, all the sides wuld be the same length Classifying Figures n a Crdinate Grid NEL

2 .4 Length Slpe y - y ( 1-1 ) + ( y - y 1 ) - 1 PQ [-30 - (-5)] + [10 - (-5)] 10 - (-5) m PQ (-5) 15-5 # The length f PQ is abut 9.15 units. - 5 QR [-5 - (-30)] + (5-10) # m QR -5 - (-30) 15 5 I decided t calculate the slpe and the length f each side f PQRS. The length f QR is abut 9.15 units. 3 5 RS [0 - (-5)] + (10-5) 10-5 m RS (-5) # The length f RS is abut 9.15 units. - 5 SP (-5-0) + (-5-10) m SP # The length f SP is abut 9.15 units. 3 5 PQ RS and QR SP PQ QR RS SP Since the ppsite sides are parallel and all the side lengths are equal, PQRS is a rhmbus. Perimeter # 4(9.15) The building lt is a rhmbus. Its perimeter measures abut m. The slpes f PQ and RS are the same, s they are parallel. The slpes f QR and SP are als the same, s they are parallel t. My length calculatins shwed that all fur sides are equal. Cmmunicatin Tip The symbl is used t replace the phrase is parallel t. I multiplied the side length by 4 t calculate the perimeter. Reflecting A. Why culd Anita nt rely cmpletely n her diagram t determine the shape f the quadrilateral? B. Why did Anita need t calculate the slpes and lengths f all the sides t determine the shape f the building lt? Eplain. NEL Chapter 97

3 APPLY the Math EXAMPLE Reasning abut lengths and slpes t classify a A has vertices at A(-1, -1), B(, 0), and C(1, 3). What type f is it? Justify yur decisin. Angelica s Slutin 4 0 A( 1, 1) - y C(1, 3) B(, 0) 4 I drew a diagram f the n grid paper. I thught that the might be issceles since AB and BC lk like they are the same length. The might als be a right since ABC lks like it might be 90. AB [ - (-1)] + [0 - (-1)] # BC (1 - ) + (3-0) (-1) + 3 # AB and BC are the same length, s ^ ABC is issceles. Slpe y - y 1-1 m AB 0 - (-1) - (-1) 1 3 m BC The slpes f AB and BC are negative reciprcals, s AB BC. This means that ^ ABC is a right. ^ ABC is an issceles right, with AB BC and ABC 90. T determine the type f, I had t knw the lengths f the sides. T check my issceles predictin, I used the distance frmula t determine the lengths f AB and BC. I runded each answer t ne decimal place. T determine if the has a right angle, I had t determine if a pair f sides are perpendicular. I did this by cmparing their slpes. I calculated the slpes f the sides that lked perpendicular, AB and BC. Cmmunicatin Tip The symbl is used t replace the phrase is perpendicular t Classifying Figures n a Crdinate Grid NEL

4 .4 EXAMPLE 3 Slving a prblem using prperties f line segments Tny is cnstructing a patterned cncrete pati that is in the shape f an issceles, as requested by his client. On his plan, the vertices f the are at P(, 1), Q(5, 7), and R(8, 4). Each unit represents 1 m. a) Cnfirm that the plan shws an issceles. b) Calculate the area f the pati. Tny s Slutin a) 8 y Q(5, 7) R(8, 4) P(, 1) I made a sketch f the. It lks issceles since PQ and RP appear t be the same length. Length ( - 1 ) + ( y - y 1 ) PQ (5 - ) + (7-1) # I used the distance frmula t calculate the lengths f the sides f the. I saw that PQ is the same length as RP, s the is issceles. b) RP ( - 8) + (1-4) # QR (8-5) + (4-7) # Since PQ RP, the is issceles y Q(5, 7) P(, 1) M R(8, 4) I knew that I needed the lengths f the base and height t calculate the area f the, since base * height area. I remembered that, in an issceles, the median frm the verte where the tw equal sides meet is perpendicular t the side that is ppsite this verte. PM is the height f the and QR is its base. NEL Chapter 99

Midpint f QR is M a 1 +, y 1 + y b a 5 + 8, 7 + 4 b (6.5, 5.5) I calculated M, the midpint f QR, s I culd use it t determine the length f PM. PM (6.5 - ) + (5.5-1) 0.5 + 0.5 # 40.5 6.

5 Midpint f QR is M a 1 +, y 1 + y b a 5 + 8, b (6.5, 5.5) I calculated M, the midpint f QR, s I culd use it t determine the length f PM. PM (6.5 - ) + (5.5-1) # Area f ^ PQR QR * PM 18 * The triangular pati has an area f 13.5 m. I used the distance frmula t calculate the length f PM. I already knew that the length f QR is 18. PM is the height f the, and QR is the base. I calculated the area f the using the eact values t minimize the runding errr. I checked my calculatins by pltting the vertices f the using dynamic gemetry sftware. Then I cnstructed the and its interir. I measured the lengths f the sides and determined the area. The scale in this sketch is 1 unit 1 cm instead f 1 unit 1 m. Since the numbers were the same, hwever, I knew that my calculatins were crrect. Tech Supprt D nt change the scale f the dynamic gemetry sftware grid, because this will make the unit length different frm 1 cm Classifying Figures n a Crdinate Grid NEL

6 .4 In Summary Key Idea When a gemetric figure is drawn n a crdinate grid, the crdinates f its vertices can be used t calculate the slpes and lengths f the line segments, as well as the crdinates f the midpints. Need t Knw Triangles and quadrilaterals can be classified by the relatinships between their sides and their interir angles. Triangles equilateral issceles scalene right issceles right Quadrilaterals parallelgram rectangle rhmbus square irregular quadrilateral trapezid issceles trapezid kite T slve a prblem that invlves a gemetric figure, it is a gd idea t start by drawing a diagram f the situatin n a crdinate grid. Parallel lines have the same slpe. Perpendicular lines have slpes that are negative reciprcals. CHECK Yur Understanding Rund all answers t tw decimal places, where necessary. 1. Shw that the line segment jining pints P(1, 4) and Q(5, 5) is parallel t the line segment jining pints R(3, -4) and S(7, -3).. Shw that TU, T(-1, 7) and U(3, 5), is perpendicular t VW, V(-4, 1) and W(-1, 7). NEL Chapter 101

7 3. The sides f quadrilateral ABCD have the fllwing slpes. Side AB BC CD AD 1 1 Slpe What types f quadrilateral culd ABCD be? What ther infrmatin is needed t determine the eact type f quadrilateral? 4. ^DEF has vertices at D(-3, -4), E(-, 4), and F(5, -5). a) Shw that ^DEF is issceles. b) Determine the length f the median frm verte D. c) Shw that this median is perpendicular t EF. PRACTISING 5. The lengths f the sides in a quadrilateral are PQ 4.5 units, QR 4.5 units, RS 4.5 units, and SP 4.5 units. What types f quadrilateral culd PQRS be? What ther infrmatin is needed t determine the eact type f quadrilateral? 6. The fllwing pints are the vertices f s. Predict whether each is scalene, issceles, r equilateral. Then draw the n a crdinate grid and calculate each side length t check yur predictin. a) A(3, 3), B(-1, ), C(0, -) c) D(, -3), E(-, -4), F(6, -6) b) G(-1, 3), H(-, -), I(, 0) d) J(, 5), K(5, -), L(-1, -) 7. P(-7, 1), Q(-8, 4), and R(-1, 3) are the vertices f a. Shw that ^ PQR is a right. 8. A has vertices at L(-7, 0), M(, 1), and N(-3, 5). Verify that it is a right issceles. 9. a) Hw can yu use the distance frmula t decide whether pints P(-, -3), Q(4, 1), and R(, 4) frm a right? Justify yur answer. b) Withut drawing any diagrams, eplain which sets f pints are the vertices f right s. i) S(-, ), T(-1, ), U(7, 0) ii) X(3, ), Y(1, -), Z(-3, 6) iii) A(5, 5), B(3, 8), C(8, 7) 10. A quadrilateral has vertices at W(-3, ), X(, 4), Y(6, -1), and Z(1, -3). a) Determine the length and slpe f each side f the quadrilateral. b) Based n yur calculatins fr part a), what type f quadrilateral is WXYZ? Eplain. c) Determine the difference in the lengths f the tw diagnals f WXYZ Classifying Figures n a Crdinate Grid NEL

8 11. A plygn is defined by pints R(-5, 1), S(5, 3), T(, -1), and U(-8, -3). Shw that the plygn is a parallelgram. 1. A quadrilateral has vertices at A(-, 3), B(-, -), C(, 1), and D(, 6). Shw that the quadrilateral is a rhmbus. 13. a) Shw that EFGH, with vertices at E(-, 3), F(, 1), G(0, -3), K and H(-4, -1), is a square. b) Shw that the diagnals f EFGH are perpendicular t each ther. 14. The vertices f quadrilateral PQRS are at P(0, -5), Q(-9, ), R(-5, 8), and S(4, ). Shw that PQRS is nt a rectangle. 15. A square is a special type f rectangle. A square is als a special type C f rhmbus. Hw wuld yu apply these descriptins f a square when using the crdinates f the vertices f a quadrilateral t determine the type f quadrilateral? Include eamples in yur eplanatin. 16. Determine the type f quadrilateral described by each set f vertices. Give reasns fr yur answers. a) J(-5, ), K(-1, 3), L(-, -1), M(-6, -) b) E(-5, -4), F(-5, 1), G(7, 4), H(7, -1) c) D(-1, 3), E(6, 4), F(4, -1), G(-3, -) d) P(-5, 1), Q(3, 3), R(4, -1), S(-4, -3) 17. A surveyr is marking the crners f a building lt. If the crners have A crdinates A(-5, 4), B(4, 9), C(9, 0), and D(0, -5), what shape is the building lt? Include yur calculatins in yur answer. 18. Pints P(4, 1), Q(9, 14), and R(13, 4) are three vertices f a rectangle. T a) Determine the crdinates f the furth verte, S. b) Briefly describe hw yu fund the crdinates f S. c) Predict whether the lengths f the diagnals f rectangle PQRS are the same length. Check yur predictin. 19. Suppse that yu knw the crdinates f the vertices f a quadrilateral. What calculatins wuld help yu determine if the quadrilateral is a special type, such as a parallelgram, rectangle, rhmbus, r square? Hw wuld yu use the crdinates f the vertices in yur calculatins? Organize yur thughts in a flw chart..4 Etending 0. a) Shw that the midpints f any pair f sides in a are tw f the vertices f anther, which has dimensins that are eactly ne-half the dimensins f the riginal and a side that is parallel t a side in the riginal. b) Shw that the midpints f the sides in any quadrilateral are the vertices f a parallelgram. NEL Chapter 103

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