Geometry 1-6 and 1-7 Notes August 22, 2016 or August 23, 2016

1-6 The Coordinate Plane Learning Objectives Find the distance between two points on the coordinate plane Find the midpoint of a segment on the coordinate plane If two points are on a horizontal or vertical line, we can count to find the distance between (or use the ruler postulate) The distance is 2 units

1-6 The Coordinate Plane You can describe a point by an ordered pair (x,y) Called the coordinates of the point When two points are not on a horizontal or vertical line, we need to use the coordinates (x1,y1) (x2,y2) Point A has coordinates (x1,y1). We will use this as our first point (1,1) so x=1 and y=1 Point B has coordinates (x2,y2)=(3,3) so x=3 and y=3 B (3,3) d A (1,1)

1-6 The Coordinate Plane The Distance Formula Example The distance d between two points A (x1,y1) and B (x2,y2) is: Example Find the distance between A(1,1) and B(3,3) x1= 1, y1= 1 and x2 = 3, y2 = 3 𝑑= (3−1) 2 + (3−1) 2 𝑑= (2) 2 + (2) 2 𝑑= 4+4 𝑑= 8 = 2.8 (rounded)

1-6 The Coordinate Plane The Distance Formula Example The distance d between two points A (x1,y1) and B (x2,y2) is: Example Find the distance between R(-1,4) and S(2,3) x1= -1, y1= 4 and x2 = 2, y2 = 3 𝑑= (2−(−1) 2 + (3−4) 2 𝑑= (3) 2 + (−1) 2 𝑑= 9+1 𝑑= 10 = 3.2

1-6 The Coordinate Plane To find the midpoint, find the average of the endpoints coordinates. Our midpoint has an x-value that is halfway between the x’s and a y-value that is halfway between the y’s. Midpoint Formula The coordinates of the midpoint M of 𝐴𝐵 with endpoints A (x1,y1) and B (x2,y2) are the following: 𝑀=( x1+ x2 2 , y1 + y2 2 )

1-6 The Coordinate Plane 𝑀=( x1+ x2 2 , y1 + y2 2 ) Example 𝐴𝐵 has endpoints A(3,5) and B(0,-3). Find the coordinates of its midpoint M. A (3,5) (x1,y1) B (0,-3)(x2,y2) 𝑀=( 3 + 0 2 , 5 + −3 2 ) 𝑀=( 3 2 , 2 2 ) 𝑀= (1 1 2 ,1) A M B

1-7 Perimeter, Circumference, and Area Learning Objectives Find perimeter and circumference Find area of rectangles, circles, and irregular shapes Understand and use Postulates 1-9 and 1-10 Perimeter – is the distance around a figure Area – is the space inside a figure Perimeter = h + b + h + b or P = 2h + 2b P = 2(2cm) + 2(5cm) = 14cm Area = bh A = (2cm)(5cm) = 10cm b 5cm h h 2cm b

1-7 Perimeter, Circumference, and Area Squares Squares are rectangles with the same side length, s Perimeter: P = 4s Area: A = 𝑠 2 Circles Circumference: C = 𝜋𝑑 or C = 2𝜋r Area: A = 𝜋 𝑟 2 𝜋 is irrational and goes on forever so we can never have an exact answer with 𝜋. For an approximate answer, use 𝜋 = 3.14 For a rounded answer, use the 𝜋 on your calculator For an exact answer, leave 𝜋 in the answer D S R S S Radius = r Diameter = d S

1-7 Perimeter, Circumference, and Area Example Find the circumference C = 𝜋𝑑 C = 3𝜋 use this for an exact answer C ≈ 3 3.14 = 9.42 use this for an approximate C = 3·𝜋 in calculator = 9.4247779608 ≈ 9.42 rounded 3m

1-7 Perimeter, Circumference, and Area Example: Finding Perimeter in the Coordinate Plane Find the perimeter of ABC with points A(-3,-1), B(2,-1), C(2,5) First: Sketch out shape AB = 2−(−3) = 5 BC = 5− −1 =6 𝐴𝐶= (2−(−3)) 2 + (5−(−1)) 2 = 5 2 + 6 2 = 25+36 = 61 ≈ 7.8 AB + BC + AC = 5 + 6 + 7.8 = 18.8 units Perimeter of ABC is 18.8 units C (2,5) Use the Ruler Postulate Use the Distant Formula A (-3,-1) B (2,-1)

1-7 Perimeter, Circumference, and Area Postulate 1-9 If two figures are congruent, then their areas are equal Postulate 1-10 The area of a region is the sum of the areas of its non-overlapping parts *Postulate 1-10 tell us that to find the area of an irregular shape, we just need to break it into shapes we can find the area of, these shapes we can find the area of, if added together, will equal the irregular shape.

Homework 1-6 pg. 46 #1-6, 11, 12, 18-23 even, 37, 38, 53, 55 1-7 pg. 55 #1, 3, 6, 7, 10, 11, 14-16, 20, 24, 28, 29, 32, 39, 40, 50, 59, 70