The Distance and The Distance and Midpoint Formulas!Midpoint Formulas!

To be used when you want to find To be used when you want to find the the distancedistance between two points between two points

or the or the midpointmidpoint between two between two pointspoints

You have learned…You have learned… 11

The PythagoreanThe Pythagoreana a c c TheoremTheorem

aa22 + b + b22 = c = c22

b 2b 2When we are finding c we are really When we are finding c we are really

finding the distance between angles 1 finding the distance between angles 1 and 2and 2

If we solve for c we get c = If we solve for c we get c = √(a√(a22 + b + b22))This is how the distance formula is derived This is how the distance formula is derived

– it is useful when we know coordinates– it is useful when we know coordinates

The Distance Formula!The Distance Formula!

The distance formula isThe distance formula is

d = d = √((y√((y22 – y – y11))22 + (x + (x22 – x – x11))22))

And is used to find the distance between And is used to find the distance between two points on the coordinate planetwo points on the coordinate plane

Let’s practice one!Let’s practice one!

Example:Example:

What is the distance between (2, -6) and (-What is the distance between (2, -6) and (-3, 6)?3, 6)?

First, identify x1, y1, x2 and y2First, identify x1, y1, x2 and y2

xx11 = 2 = 2 yy11 = -6 = -6

xx22 = -3 = -3 yy22 = 6 = 6

Now, we use the formula:Now, we use the formula:

d = d = √((6 – -6)√((6 – -6)22 + (-3 – 2) + (-3 – 2)22) = √(12) = √(1222 + (-5 + (-52)2)))

d = 13d = 13

The Midpoint Formula!The Midpoint Formula!The midpoint formula is used to find the The midpoint formula is used to find the

coordinate that is the exact midpoint coordinate that is the exact midpoint between two other coordinatesbetween two other coordinates

The x-coordinate of the midpoint is found byThe x-coordinate of the midpoint is found by

(x(x22 + x + x11)/2)/2The y-coordinate of the midpoint is found byThe y-coordinate of the midpoint is found by

(y(y22 + y + y11)/2)/2So, the coordinate of the midpoint is:So, the coordinate of the midpoint is:

(x(x22 + x + x11)/2, (y)/2, (y22 + y + y11)/2)/2

Example:Example:What is the coordinate of the midpoint What is the coordinate of the midpoint

between (1, 2) and (-5, 6)?between (1, 2) and (-5, 6)?

Again, identify x1, y1, x2 and y2Again, identify x1, y1, x2 and y2

xx11 = 1 = 1 yy11 = 2 = 2

xx22 = -5 = -5 yy22 = 6 = 6

Now use the formula:Now use the formula:

x-coordinate: (1 + -5)/2 = -2x-coordinate: (1 + -5)/2 = -2

y-coordinate: (2 + 6)/2 = 4y-coordinate: (2 + 6)/2 = 4

So, the midpoint is located at the coordinate (-So, the midpoint is located at the coordinate (-2, 4)2, 4)

Another ExampleAnother Example

On the coordinate plane, it is given that On the coordinate plane, it is given that the midpoint of points A and B is (5, the midpoint of points A and B is (5, 7). If point A is located at (-1, 2), 7). If point A is located at (-1, 2), where is point B located?where is point B located?

In this case, we know the midpoint and In this case, we know the midpoint and the coordinate of point A. In a sense, the coordinate of point A. In a sense, we need to work backwards. Let’s we need to work backwards. Let’s define what we have:define what we have:

xx11 = -1 = -1 yy11 = 2 = 2

xx22 = ? = ? yy22 = ? = ?

So we know…So we know…

(-1 + x(-1 + x22)/2 = 5)/2 = 5 (2 + y(2 + y22)/2 = 7)/2 = 7

-1 + x-1 + x22 = 10 = 10 (2 + y(2 + y22) = 14) = 14

xx22 = 11 = 11 yy22 = 12 = 12

So the coordinate of point B is (11, 12)So the coordinate of point B is (11, 12)