10.5 Perimeter and Area on the Coordinate Plane

Transcription

1 Name lass ate 1.5 Perimeter and rea on the oordinate Plane ssential Question: How do ou find the perimeter and area of polgons in the coordinate plane? Resource Locker plore inding Perimeters of igures on the oordinate Plane Recall that the perimeter of a polgon is the sum of the lengths of the polgon s sides. You can use the istance ormula to find perimeters of polgons in a coordinate plane. ollow these steps to find the perimeter of a pentagon with vertices ( 1, ), (, ), (3, ), ( 1, ), and (, 1). Round to the nearest tenth. Plot the points. Then use a straightedge to draw the pentagon that is determined b the points. re there an sides for which ou do not need to use the istance ormula? plain, and give their length(s) Use the istance ormula to find the remaining side lengths. Houghton Mifflin Harcourt Publishing ompan ind the sum of the side lengths. Reflect 1. plain how ou can find the perimeter of a rectangle to check that our answer is reasonable. Module 1 59 Lesson 5

2 plain 1 inding reas of igures on the oordinate Plane You can use area formulas together with the istance ormula to determine areas of figures such as triangles, rectangles and parallelograms. triangle = bh h b rhombus = d 1 d h rectangle = bh b kite = d 1 d parallelogram = bh = b h trapezoid ( b 1 + b ) h d 1 d d 1 d b 1 h b ample 1 ind the area of each figure. Step 1 ind the coordinates of the vertices of. (-, -), (-, ), (5, 1) Step hoose a base for which ou can easil find the height of the triangle. _ Use as the base. segment from the opposite verte,, to point (-1, -1) appears to be perpendicular to the base _. Use slopes to check. slope of _ 1 - (-) = _ 5 - (-) = 3 ; slope of _ = _ (-) = -3 The product of the slopes is 1 _ 3 (-3) = -1. _ is perpendicular to _, so _ is the height for the base _. ind the length of the base and the height. = (5 - (-) ) + (1 - (-) ) = Step 3 etermine the area of. rea = bh = ()() = (3 9 = 3 1 ) ( ; = 1 ) = _ 1 3 = 15 square units (-1 - (-)) + (-1 - ) = 1 Houghton Mifflin Harcourt Publishing ompan Module 1 55 Lesson 5

3 Step 1 ind the coordinates of the vertices of G. 6 (-, 6), (, 3), (, -1), G (-, ) Step G appears to be a rectangle. Use slopes to check that adjacent sides are perpendicular. G - - slope of _ - :_ = _ - (-) 6 = ; slope of _ :_ - 3 = _ = - slope of _ G :_ - = _ = ; slope of _ G : - = _ = so G is a. Step 3 ind the area of G. b = G = ( - ) + ( - ) = _ = h = G = ( - (-) ) + ( 6 - ) = _ = rea of G: = bh = ( ) ( ) = square units Reflect. In Part, is it possible to use another side of as the base? If so, what length represents the height of the triangle? Houghton Mifflin Harcourt Publishing ompan 3. iscussion In Part, wh was it necessar to find the slopes of the sides? Module Lesson 5

4 Your Turn. ind the area of quadrilateral JKLM with vertices J (, ), K (, 1), L (3, ), M ( 3, 1) plain inding reas of omposite igures composite figure is made up of simple shapes, such as triangles, rectangles, and parallelograms. To find the area of a composite figure, find the areas of the simple shapes and then use the rea ddition Postulate. You can use the rea ddition Postulate to find the area of a composite figure. rea ddition Postulate The area of a region is equal to the sum of the areas of its nonoverlapping parts. ample ind the area of each figure. Possible solution: can be divided up into a rectangle and two triangles, each with horizontal bases. area of rectangle G: = bh = ()() = (6)() = area of : = bh = ()() = (8)() = 8 area of G: = bh = _ 1 (G)(G) = _ 1 ()() = area of : = = 36 square units G Houghton Mifflin Harcourt Publishing ompan Module 1 55 Lesson 5

5 PQRST can be divided into a parallelogram and a triangle. PQT appears to be a right triangle. heck that PT _ and perpendicular: slope of _ PT : _ 1 - = _ = -3 - are P T - S Q R slope of : _- 3 = _ = -3 - PQT is a right triangle with base _ PT and height. ( -3 - ) PT = + ( 1 - ) = _ ( -3 - ) = + ( -3) = _ = _ area of PQT: = bh = _ QR TS _ since both sides are vertical. ( _ ) ( _ ) = slope of = _-1 = _ =, so QT _. Therefore, QRST is a parallelogram _ RT is an of QRST and is horizontal. ecause RT RQ, QRT is a right Houghton Mifflin Harcourt Publishing ompan triangle with base and height. Therefore, the area of QRST = (area of QRT). RT =, QR =, so the area of QRT = ( ) ( ) = 6. QRST = (area of QRT) = = 1 area of PQRST: = + = square units Reflect 5. iscussion How could ou use subtraction to find the area of a figure on the coordinate plane? Module Lesson 5

6 Your Turn 6. ind the area of the polgon b addition. 7. ind the area of polgon b subtraction. W J - - G - - Z - X - I H - Y plain 3 G Using Perimeter and rea in Problem Solving You can use perimeter and area techniques to solve problems. ample 3 Miguel is planning and costing an ornamental garden in the a shape of an irregular octagon. ach unit on the coordinate grid represents one ard. He wants to la the whole garden with turf, which costs $3.5 per square ard, and surround it with a border of decorative stones, which cost $7.95 per ard. What is the total cost of the turf and stones? H Houghton Mifflin Harcourt Publishing ompan Image redits: lena rozova/shutterstock Module 1 55 Lesson 5

7 nalze Information Identif the important information. The vertices are (, 5) ( ) ( ) ( ), 1,, 6,,,, ( 1, 3), (, 3), G ( 5, ), and H (, ). The cost of turf is $ per square ard. The cost of the ornamental stones is $ per ard. ormulate a Plan ivide the garden up into. dd up the of the smaller figures. ind the cost of turf b the total area b the cost per square ard. ind the perimeter of the garden b adding the of the sides. ind the cost of the border b the perimeter b the cost per ard. ind total cost b adding the and. Solve Houghton Mifflin Harcourt Publishing ompan ivide the garden into smaller figures. The garden can be divided into square, kite H, and parallelogram GH. ind the area of each smaller figure. area of : slope of : _ - _ = slope of :_ = - lso, = ( 1) - + ( - ) = _ and = ( - ) + ( ) - = _. G H - So is a square, with area = s = ( _ d) = d. Module Lesson 5

8 area of kite H: H = (-1 - ( ) ) ( - (-1) ) + ( - ) = + ( - ) = + = _ = + 9 = ; H = (-1 - ( ) ) + ( ) - = + 5 = = + (-3 - ) = + = ( -1) So, and. Therefore H is a kite. b = d 1 = 8, h = d = = d 1 d = area of parallelogram GH: ( ) ( ) = d and GH are both horizontal, so are parallel; ; ; slope of _ H : - - _ = _ = ; slope of : = _ = So GH is a parallelogram, with base = and height. H =. area of GH: = bh = ( d ) ( d ) = d ind the total area of the garden and the cost of turf. area of garden: = d + d + d = d cost of turf: ( d ) ( $ / d ) = $ ind the perimeter of the garden. = d, GH = d rom area calculations, = = = _ d, and = H = ( ) - G = + ( - (-3) ) = _, perimeter of garden = GH + H G = ind the cost of the stones for the border. cost of stones: ( + d) ($ per d) $ ind the total cost. + total cost: $ + $ = $ d Houghton Mifflin Harcourt Publishing ompan Module Lesson 5

9 Justif and valuate The area can be checked b subtraction: area of large rectangle = (11) (1) = 11 square units area = (11)( ) - ( )(3) - _ 1 () ( ) - _ 1 ( )( ) - (5)( ) - _ 1 ( )() - ( )(5) - ( )( ) - ( )( ) - ( )( ) G a b c i H - - h - g d e f = _ = The perimeter is approimatel the perimeter of the polgon shown: The perimeter of the polgon shown is, so the answer is reasonable. G H Houghton Mifflin Harcourt Publishing ompan Your Turn 8. designer is making a medallion in the shape of the letter L. ach unit on the coordinate grid represents an eighth of an inch, and the medallion is to be cut from a 1-in. square of metal. How much metal is wasted to make each medallion? Write our answer as a decimal U P R Q S T Module Lesson 5

10 laborate 9. reate a flowchart for the process of finding the area of the polgon G. Your flowchart should show when, and wh, the Slope and istance ormulas are used G 1. iscussion If two polgons have approimatel the same area, do the have approimatel the same perimeter? raw a picture to justif our answer. 11. ssential Question heck-in What formulas might ou need to solve problems involving the perimeter and area of triangles and quadrilaterals in the coordinate plane? Houghton Mifflin Harcourt Publishing ompan Module Lesson 5

11 valuate: Homework and Practice ind the perimeter of the figure with the given vertices. Round to the nearest tenth. 1. (, 1), (5, ), and (, 6). P (, 5), Q (-3, ), R (, -5), and S (6, ) Online Homework Hints and Help tra Practice 3. M (-3, ), N (1, ), P (, ), Q (, -1), and R (, ). (-5, 1), (, 3), (5, 1), (, -), (, -), and (-, -) ind the area of each figure. 5. J 6. P Houghton Mifflin Harcourt Publishing ompan K - - L - M - S - - R T Q Module Lesson 5

12 ind the area of each figure b addition M N - - J - K L ind the area of each figure b subtraction P Q Q R S - R S U W - V T - - Y P - X U W T V Houghton Mifflin Harcourt Publishing ompan Module 1 56 Lesson 5

13 11. encing costs $1.5 per ard, and each unit on the grid represents 5 d. How much will it cost to fence the plot of land represented b the polgon? machine component has a geometric shaped plate, represented on the coordinate grid. ach unit on the grid represents 1 cm. ach plate is punched from an 8-cm square of allo. The cost of the allo is $.3/c m, but $.8/c m can be recovered on wasted scraps of allo. What is the net cost of allo for each component? Houghton Mifflin Harcourt Publishing ompan Image redits: micheldenijs/istockphoto.com with vertices (1, 1) and (3, 5) has an area of 1 units. What is the location of the third verte? Select all that appl.. ( 5, 5). (3, 5). (, 5). (6, 1). (3, 3) Module Lesson 5

14 1. Pentagon shows the path of an obstacle course, where each unit of the coordinate plane represents 1 meters. ind the length of the course to the nearest meter. - - lgebra Graph each set of lines to form a triangle. ind the area and perimeter. 15. =, = 5, and = 16. = 5, =, and = Prove that quadrilateral JKLM with vertices J(1, 5), K(, ), L(1, ), and M(, ) is a kite, and find its area Houghton Mifflin Harcourt Publishing ompan Image redits: Steve Skjold/lam Module 1 56 Lesson 5

15 H.O.T. ocus on Higher Order Thinking 18. plain the rror Wendell is tring to prove that is a rhombus and to find its area. Identif and correct his error. (Hint: rhombus is a quadrilateral with four congruent sides.) = ( - (-) ) + (5 - ) = 5 = 5, = (6 - ) + ( - 5) = 5 = 5 = ( - 6) + (-1 - ) = 5 = 5, = - ( - (-) ) + (-1 - () ) = 5 = 5 _ So _, and therefore is a rhombus. area of : b = = 5 and h = = 5, so = bh = (5) (5) = ommunicate Mathematical Ideas Using the figure, prove that the area of a kite is half the product of its diagonals. (o not make numerical calculations.) Houghton Mifflin Harcourt Publishing ompan. Justif Reasoning Use the figure to derive the formula for the area of a trapezoid. Then use the Trapezoid Midsegment Theorem to show that the area of a trapezoid is the product of the length of its midsegment and its height. h b 1 b Module Lesson 5

16 Lesson Performance Task The coordinate plane shows the floor plan of two rooms in ritz s house. ecause he enjos paradoes, ritz has decided to entertain his friends with one b drawing lines on the floor of his tiled kitchen, on the left, and his tiled recreation room, on the right. The four sections in the kitchen are congruent to the four sections in the recreation room. ach square on the floor plan measures 1 ard on a side. 1. ind the area of each of the four sections of the kitchen. dd the four areas to find the total area of the kitchen.. ind the area of the kitchen b finding the product of the length and the width. 3. ind the area of the recreation room b finding the product of the length and the width Kitchen Rec Room escribe the parado. 5. plain the parado. Houghton Mifflin Harcourt Publishing ompan Module 1 56 Lesson 5