Chapter 3 Summary 3.1. Determining the Perimeter and Area of Rectangles and Squares on the Coordinate Plane. Example
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1 Chapter Summar Ke Terms bases of a trapezoid (.) legs of a trapezoid (.) composite figure (.5).1 Determining the Perimeter and Area of Rectangles and Squares on the Coordinate Plane The perimeter or area of a rectangle can be calculated using the distance formula or b counting the units of the figure on the coordinate plane. When using the counting method, the units of the -ais and -ais must be considered to count accuratel. Eample Determine the perimeter and area of rectangle JKLM. 00 J K M L The coordinates for the vertices of rectangle JKLM are J(10, 50), K(60, 50), L(60, 50), and M(10, 50). Because the sides of the rectangle lie on grid lines, subtraction can be used to determine the lengths. JK 5 60 (10) KL 5 50 (50) A 5 bh P 5 JK 1 KL 1 LM 1 JM (00) 5 5,000 The area of rectangle JKLM is 5,000 square units. The perimeter of rectangle JKLM is 960 units. 7
2 .1 Using Transformations to Determine the Perimeter and Area of Geometric Figures If a rigid motion is performed on a geometric figure, not onl are the pre-image and the image congruent, but both the perimeter and area of the pre-image and the image are equal. Knowing this makes solving problems with geometric figures more efficient. Eample Determine the perimeter and area of rectangle ABCD. A 0 B D A9 C B9 D9 C The vertices of rectangle ABCD are A(0, 0), B(60, 0), C(60, 60), and D(0, 60). To translate point D to the origin, translate ABCD to the right 0 units and down 60 units. The vertices of rectangle A9B9C9D9 are A9(0, 0), B9(0, 0), C9(0, 0), and D9(0, 0). Because the sides of the rectangle lie on grid lines, subtraction can be used to determine the lengths. A9D C9D P 5 A9B9 1 B9C9 1 C9D9 1 A9D The perimeter of rectangle A9B9C9D9 and, therefore, the perimeter of rectangle ABCD, is 00 units. A 5 bh 5 0(0) The area of rectangle A9B9C9D9 and, therefore, the area of rectangle ABCD, is 1600 square units. Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane
3 . Determining the Perimeter and Area of Triangles on the Coordinate Plane The formula for the area of a triangle is half the area of a rectangle. Therefore, the area of a triangle can be found b taking half of the product of the base and the height. The height of a triangle must alwas be perpendicular to the base. On the coordinate plane, the slope of the height is the negative reciprocal of the slope of the base. Eample Determine the perimeter and area of triangle JDL. J 6 P D L The vertices of triangle JDL are J(1, 6), D(7, 9), and L(, ). JD 5 ) DL 5 ) LJ 5 ) 5 (7 1) 1 (9 6) 5 ( 7) 1 ( 9) 5 (1 ) 1 (6 ) (6) 5 (7) P 5 JD 1 DL 1 LJ The perimeter of triangle JDL is approimatel 0. units. Chapter Summar 9
4 To determine the area of the triangle, first determine the height of triangle JDL. Slope of JD : m Slope of PL : m 5 Equation of JD : ( ) 5 m( ) Equation of PL : ( ) 5 m( ) 6 1) 5 ) Intersection of JD and PL, or P: (5.) The coordinates of P are (5.,.). Height of triangle JDL: PL 5 ) 5 ( 5.) 1 (.) 5 (.6) 1 (5.) Area of triangle JDL: A 5 1 bh (JD)(PL) ( 5 )(. ) ( 169 ) The area of triangle JDL is 19.5 square units. 90 Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane
5 . Doubling the Area of a Triangle To double the area of a triangle, onl the length of the base or the height of the triangle need to be doubled. If both the length of the base and the height are doubled, the area will quadruple. Eample Double the area of triangle ABC b manipulating the height. C9 C 6 Area of ABC Area of ABC9 B A A 5 1 bh A 5 1 bh (5)() 5 1 (5)() B doubling the height, the area of triangle ABC9 is double the area of triangle ABC.. Determining the Perimeter and Area of Parallelograms on the Coordinate Plane The formula for calculating the area of a parallelogram is the same as the formula for calculating the area of a rectangle: A 5 bh. The height of a parallelogram is the length of a perpendicular line segment from the base to a verte opposite the base. Eample Determine the perimeter and area of parallelogram WXYZ. 6 Z W A X Y Chapter Summar 91
6 The vertices of parallelogram WXYZ are W(, 5), X(, ), Y(, 5), and Z(, 7). WX 5 ) YZ 5 ) 5 ( ()) 1 ( (5)) 5 ( ) 1 (7 (5)) (6) 1 () WZ 5 ) XY 5 ) 5 ( ()) 1 (7 (5)) 5 ( ) 1 (5 ()) 5 (1) 1 () 5 (1) 1 () P 5 WX 1 XY 1 YZ 1 WZ The perimeter of parallelogram WXYZ is approimatel 17.1 units. To determine the area of parallelogram WXYZ, first calculate the height, AY. Slope of base WX : m (5) () 5 6 Slope of height AY : m 5 Equation of base WX : ( ) 5 m( ) Equation of height AY : ( ) 5 m( ) ( ()) ) ( (5)) 5 ) Intersection of WX and AY, or A: ( 1 1 ) Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane
7 The coordinates of point A are ( 1 1, 1 ). AY 5 ) 5 ( 1 1 ) 1 ( 5 ( 1 ) ) ( 1 ) 1 ( 1 1 ) Area of parallelogram WXYZ: A 5 bh A 5 10 (.5 ) A 5 10 The area of parallelogram WXYZ is 10 square units.. Doubling the Area of a Parallelogram To double the area of a parallelogram, onl the length of the bases or the height of the parallelogram needs to be doubled. If both the length of the bases and the height are doubled, the area will quadruple. Eample Double the area of parallelogram PQRS b manipulating the length of the bases. P Q S R S9 R9 0 6 Area of PQRS Area of PQR9S9 A 5 bh A 5 bh 5 (6)() 5 (1)() B doubling the length of the bases, the area of parallelogram PQR9S9 is double the area of parallelogram PQRS. Chapter Summar 9
8 . Determining the Perimeter and Area of Trapezoids on the Coordinate Plane A trapezoid is a quadrilateral that has eactl one pair of parallel sides. The parallel sides are known as the bases of the trapezoid, and the non-parallel sides are called the legs of the ) h, trapezoid. The area of a trapezoid can be calculated b using the formula A 5 ( b 1 1 b where b 1 and b are the bases of the trapezoid and h is a perpendicular segment that connects the two bases. Eample Determine the perimeter and area of trapezoid GAME. G 16 1 A 16 1 E 1 16 M The coordinates of the vertices of trapezoid GAME are G(, 1), A(, 1), M(, 0), and E(, 6). GA 5 ) ME 5 ) 5 ( ()) 1 (1 1) 5 (() ) 1 ((6) 0) (6) 5 (6) 1 (6) EG 5 1 (6) AM P 5 GA 1 AM 1 ME 1 EG The perimeter of trapezoid GAME is approimatel 5.0 units. 9 Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane
9 The height of trapezoid GAME is 6 units. ) h A 5 ( b 1 1 b 5 ( 1 1 ) (6) 5 10 The area of trapezoid GAME is 10 square units..5 Determining the Perimeter and Area of Composite Figures on the Coordinate Plane A composite figure is a figure that is formed b combining different shapes. The area of a composite figure can be calculated b drawing line segments on the figure to divide it into familiar shapes and determining the total area of those shapes. Eample Determine the perimeter and area of the composite figure. P 6 T S H 6 B 0 6 G 6 R The coordinates of the vertices of this composite figure are P(, 9), T(, 6), S(5, 6), B(5, 1), R(, 5), G(, 5), and H(, 1). TS 5, SB 5 5, RG 5 5, HP 5 PT 5 ) BR 5 ) GH 5 ) 5 ( ()) 1 (6 9) 5 ( 5) 1 (5 1) 5 ( ()) 1 (1 (5)) () 5 () 1 (6) 5 () 1 (6) P 5 PT 1 TS 1 SB 1 BR 1 RG 1 GH 1 HP Chapter Summar 95
10 The perimeter of the composite figure PTSBRGH is approimatel 0. units. The area of the figure is the sum of the triangle, rectangle, and trapezoid formed b the dotted lines. Area of triangle: Area of rectangle: Area of trapezoid: A 5 1 ( bh A 5 bh A 5 b 1 b 1 ) h (6)() 5 9(5) 5 ( ) (6) The area of composite figure: A The area of the composite figure PTSBRGH is 96 square units. 96 Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane
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