The Distance Formula & The Midpoint Formula

Transcription

1 The & The Professor Tim Busken Mathematics Department Januar 14, 2015

2 Theorem ( : 1 dimension) If a and b are real numbers, then the distance between them on a number line is a b. a b

3 : 2 dimensions Consider the points A( 1, 1 ) and B( 2, 2 ) in the figure below. Let d be the distance between points A and B (the HYPOTENUSE LENGTH of the right triangle). Since A and C lie on a horizontal line, the distance between them is 2 1. Likewise, CB = 2 1. B( 2, 2 ) d 2 1 A( 1, 1 ) 2 1 C( 2, 1 )

4 B( 2, 2 ) d 2 1 A( 1, 1 ) 2 1 C( 2, 1 ) Since the sum of the squares of the legs of a right triangle is equal to the square of the hpotenuse (Pth. thm), then from the diagram d 2 =

5 Theorem ( : 2 dimensions) The distance d between the points ( 1, 1 ) and ( 2, 2 ) is given b the formula d = ( 2 1 ) 2 +( 2 1 ) 2 B( 2, 2 ) d 2 1 A( 1, 1 ) 2 1 C( 2, 1 )

6 Eample: Find the eact distance between the points (5, 3) and ( 1, 6) Solution Let ( 1, 1 ) = (5, 3) and ( 2, 2 ) = ( 1, 6). (5, 3) ( 1, 6) Figure : d Then d = ( 2 1 ) 2 +( 2 1 ) 2 = ( 1 5) 2 +( 6 ( 3)) 2 = ( 6) 2 +( 3) 2 = 36+9 = 45 = 3 5

7 Theorem ( : 1 dimension) If a and b are real numbers, then the midpoint between them on a number line is a+b 2. a a+b 2 b

8 Theorem ( : 2 dimensions) Suppose ( 1, 1 ) and ( 2, 2 ) are an two points in two-dimensional space. Then the midpoint of the line segment that joins them is: ( (2 + 1 ) m = 2, ( ) 2 ). ( ) , ( 2, 2 ) ( 1, 1 ) midpoint

9 Eample: Find the midpoint between the points (5, 3) and ( 1, 6) Solution Let ( 1, 1 ) = (5, 3) and ( 2, 2 ) = ( 1, 6). (2, 9 2 ) (5, 3) ( 1, 6) Figure : midpoint Then ( (2 + 1 ) m =, ( ) ) 2 2 ( 5+( 1) =, 3+( 6) ) 2 2 ( ) 4 = 2, 9 2 ( = 2, 9 ) 2

10 The Circle An ordered pair is a solution to an equation in two variables if the equation is correct when the variables are replaced b the coordinates of the ordered pair.

11 The Circle An ordered pair is a solution to an equation in two variables if the equation is correct when the variables are replaced b the coordinates of the ordered pair. The solution set to an equation in two variables is the set of all ordered pairs that satisf the equation.

12 The Circle An ordered pair is a solution to an equation in two variables if the equation is correct when the variables are replaced b the coordinates of the ordered pair. The solution set to an equation in two variables is the set of all ordered pairs that satisf the equation. The graph of (the solution set to) an equation in two variables is a two-dimensional geometric object that gives us a visual image of an algebraic object.

13 The Circle An ordered pair is a solution to an equation in two variables if the equation is correct when the variables are replaced b the coordinates of the ordered pair. The solution set to an equation in two variables is the set of all ordered pairs that satisf the equation. The graph of (the solution set to) an equation in two variables is a two-dimensional geometric object that gives us a visual image of an algebraic object. Definition (Circle) A circle is defined b the set of all points in the plane that lie a fied distance from a given point (the center). The fied distance is called the radius, and the given point is the center.

14 The distance formula can be used to write an equation for a circle with center (h, k) and radius r for r> 0. (,) r (h,k)

15 A point (, ) is on the circle if and onl if it satisfies the equation ( h)2 +( k) 2 = r. (, ) (h, k) r k h

16 Since both sides of the equation (previous slide) are positive, we can square each side to get the standard form for the equation of a circle. r (,) (h,k)

17 Since both sides of the equation (previous slide) are positive, we can square each side to get the standard form for the equation of a circle. (,) r (h,k) Theorem (Equation for a Circle in Standard Form) The equation for a circle with center (h, k) and radius r (where r> 0) is ( h) 2 +( k) 2 = r 2 A circle centered at the origin has equation = r 2.

18 Eample: Sketch the graph of the equation ( 1) 2 +( + 2) 2 = 3 (,). r (h,k)