Distance Between Two Points Where the Distance Formula
Distance Between Two Points Where the ‘Distance Formula’ comes from and how to use it.
Background Suppose I gave you two points on the Cartesian Plane. N (-4, 3) ? P (4, -3) If I asked you to find the distance between these two points, how would you go about finding it?
Background Let’s use some grade 9 math, that you all know and love, to solve for the distance between these points. N (-4, 3) P (4, -3)
Background �Remember Pythagoras? �If you have a right angled triangle, then the sides have a certain relationship: N a 2 + b 2 = c 2 (-4, 3) where a & b are the two (4, -3) P sides touching the right angle, and c is the longest side and that is across from the right angle.
Background N 8 6 10 � a 2 P + b 2 = c 2 � 8 2 + 6 2 = c 2 � 64 + 36 = c 2 � 100 = c 2 2 � = � 10 = c the distance between points N & P is 10 units.
So what happened? ? ? Since we’re talking about distance, instead of using c, we will use d. (x 1, y 1 ) N(-4, 3) 8 d 2 = a 2 + b 2 d 2 = 8 2 + 62 d 2 = (4 -(-4))2 + (-3 -(3))2 d 2 = (x 2 -x 1)2 + (y 2 -y 1)2 d = 6 10 P (4, -3) (x 2, y 2)
Example Find the perimeter of this triangle G F H
Example Let’s start with the blue side. . G (3, 5) (x 1, y 1) d 2 = a 2 + b 2 d 2 = 32 + 102 d 2 = 9 + 100 F d 2 = 109 d = H (0, -5) (x 2, y 2) d = 10. 44
Example Let’s continue with the blue side. . G (3, 5) (x 1, y 1) d 2 = a 2 + b 2 d = F d = H (0, -5) (x 2, y 2) d = 10. 44
Example Next is the gold side. . G d 2 = a 2 + b 2 d 2 = 4 2 + 32 d 2 = 16 + 9 F (-4, -2) (x 1, y 1) d 2 = 25 d = H (0, -5) (x 2, y 2) d = 5
Example Continue with the gold side. . G d 2 = a 2 + b 2 d = F (-4, -2) (x 1, y 1) d = H (0, -5) (x 2, y 2) d = 5
Example Lastly the green side. . G (3, 5) (x 2, y 2) d 2 = a 2 + b 2 d 2 = 7 2 + 72 d 2 = 49 + 49 F (-4, -2) (x 1, y 1) d 2 = 98 H d = 9. 9
Example Lastly, let’s continue the green side. . G (3, 5) (x 2, y 2) d 2 = a 2 + b 2 d = F (-4, -2) (x 1, y 1) d = H d = 9. 9
Example �So the perimeter of our triangle is: �Perimeter = 10. 44 + 5 + 9. 9 = 25. 34
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