Notes on the Distance Formula - Page One

Notes on the Distance Formula - Page One Name______

Suppose one wants to find the 6 6 distance from point P to point Q P P 4 4 (see left). c b 2 2 Q a Q

-5 5 As previously learned, a right -5 5

-2 triangle can be formed, and the -2

-4 Pythagorean Theorem can be used to -4

-6 find this distance (see right). -6

4 P By using a simple counting method, one can find the values c 2 of a and b. 2 5 Q -5 5 In this example, a = 5 , and b = 2 .

However, consider once again the original points, with coordinates 6 of (1, 4) and (6, 2). (1, 4) 4

2 In actuality, one can find the value of a by simply finding how far (6, 2) apart the points are horizontally. As learned previously, horizontal -5 5 distance is just the difference in the x -coordinates. -2

If one subtracts the x-coordinates to find a, the result is the same: -6 a =6 - 1 = 5

Similarly, the value of b can be found by finding how far apart the points are vertically. Vertical distance is the difference in the y -coordinates.

When the y-coordinates are subtracted to get b, the result is the same: b =4 - 2 = 2 .

Therefore, a is really just the difference in the x-coordinates of the two points, and b is really just the difference in the y-coordinates.

Remember: c is the distance that we are trying to find (the distance between the two points). Let's substitute all of these ideas into the Pythagorean Theorem: a2+ b 2 = c 2

c2= a 2 + b 2

distance2= difference in x-coordinates 2 + difference in y-coordinates 2 ------So, the formula above describes how to find the distance squared. The formula will just find the distance if we take the square root of both sides.

distance= difference in x-coordinates2 + difference in y-coordinates 2

2 2 distance= (x2 - x 1 ) + ( y 2 - y 1 )

This is known as the Distance Formula.

This formula can be used to find the distance between any two points on the coordinate plane. ------Find the distance between two points whose coordinates are (7, 2) and (15, 8).

(7, 2) and (15, 8 ) First, the points will be appropriately labeled: . (x1 , y 1 ) ( x 2 , y 2 )

Now, the values will be substituted into the formula:

2 2 distance= (x2 - x 1 ) + ( y 2 - y 1 ) =(15 - 7)2 + ( 8 - 2 ) 2 =(8)2 + (6) 2 =64 + 36 = 100 = 10 units ------Find the distance between two points whose coordinates are (20, 7) and (24, 10).

2 2 distance= (x2 - x 1 ) + ( y 2 - y 1 )

Notes on the Distance Formula - Page Two Name______Find the distance between two points whose coordinates are (-3, -7) and (-1, -4). Express your answer as a decimal rounded to the nearest tenth.

2 2 x y x y distance= (x2 - x 1 ) + ( y 2 - y 1 ) 1 1 2 2 =( - 1 - - 3)2 + ( - 4 - - 7) 2 =(2)2 + (3) 2 =4 + 9 = 13 3.6 units ------Find the distance between two points whose coordinates are (-5, 1) and (-14, -6). Express your answer as a decimal rounded to the nearest tenth.

------The distance from a point to a line is defined as the length of the perpendicular segment from the point to the line.

6 For example, suppose one wants 6

line m the distance from point A to line m 4 4

A line m (see left). A 2 2 4 -5 5 A perpendicular segment would -5 5 -2 be drawn from point A to line m -2 -4 (see right). The length of this -4

-6 segment is 4 units. -6

Thus, the distance from point A to line m is 4 units. ------

4 B 2 How far away is point B from line n on the coordinate

-5 5 plane to the left?

-4 line n

Homework on Distance Formula Name______Find the distance between the two points with the given coordinates. Round all decimal answers to the nearest tenth.

1. (6, 2) and (0, 0) 2. (6, 8) and (30, 1) 3. (-2, 18) and (3, 6)

4. (-1, 0) and (0, -1) 5. (-5, -17) and (-9, -19) 6. (-2, 7) and (-7, -3)

------Land mines have been placed in a field. The layout of the field and the 1 6 locations of the mines have been placed on a coordinate plane, and each 2 4 unit represents 10 yards. Round answers to the nearest yard.

4 -5 3 5 7. How far apart is land mine 2 from land mine 3?

5 -4 8. How far apart is land mine 1 from land mine 6? 6

-6 9. Add land mine 7 to the coordinate plane so that it is 40 yards from land mine 5.

10. If the entire field is pictured, what is the length of the field?

------11. In the movie, Liar, Liar, Jim Carrey’s character runs 6 to catch his son’s plane on the runway. If the point J on the coordinate plane represents Jim Carrey, and the 4 J line R represents the runway, what is the distance in units 2 from Jim Carrey to the runway? -5 5

-2 If each unit represents 30 feet, what is the distance in feet from Jim Carrey to the runway? -4 line R -6

------12. Suppose point A has coordinates of (3, 4), and point B has coordinates of (10, 4). Phillip says, "The line that passes thru point A and point B must be a horizontal line." How does he know that? ------1. 6.3 units 2. 25 units 3. 13 units 4. 1.4 units 5. 4.5 units 6. 11.2 units 7. about 57 yards 8. about 108 yards 9. to be discussed 10. 140 yards 11. 7 units = 210 feet 12. Since the two points have the same y-coordinate (4), they must lie on the same horizontal line.