# FINDING THE DISTANCE BETWEEN TWO POINTS

The Pythagorean Theorem can be used to find the distance between any two points (x1, y1) and (x2, y2) in the coordinate plane. The resulting expression is called the Distance Formula.

In a coordinate plane, the distance d between two points (x1, y1) and (x2, y2) is ## Using the Pythagorean Theorem to derive the Distance Formula

Step 1 :

To find the distance between points P(x1, y1) and Q(x2, y2)draw segment PQ and label its length d. Then draw horizontal segment PR and vertical segment QR . Label the lengths of these segments a and b. PQR is a right triangle with hypotenuse PQ  =  d.

Step 2 :

Since PR is a horizontal segment, its length, a, is the difference between its x-coordinates. Therefore,

a  =  x2 - x1

Step 3 :

Since QR is a horizontal segment, its length, b, is the difference between its y-coordinates. Therefore,

b  =  y2 - y1

Step 4 :

Use the Pythagorean Theorem to find d, the length of segment PQ. Substitute the expressions from step 2 and step 3 for a and b.

d2  =  a2 + b2

Thus, we have ## Reflect

Why are the coordinates of point R the ordered pair (x₂, y₁) ?

Since R lies on the same vertical line as Q, their x-coordinates are the same. Since R lies on the same horizontal line as P, their y-coordinates are the same.

## Practice Problems

Problem 1 :

Find the distance between the points A(-12, 3) and B(2, 5).

Solution:

Step 1 :

Write the formula to find the distance between the two points A(-12, 3) and B(2, 5).

AB  =  √[(x2 - x1)2(y2 - y1)2]

Step 2 :

Substitute x1 = -12, y1 = 3, x2 = 2, and y2 = 5.

AB  =  √[(2 + 12)2 + (5 - 3)2]

AB  =  √(142 + 22)

AB  =  √(196 + 4)

AB  =  √200

AB  =  √(2 x 10 x 10)

AB  =  10√2 units

Problem 2 :

Find the distance between the points P(-2, -3) and Q(6, -5).

Solution:

Step 1 :

Write the formula to find the distance between the two points P(-2, -3) and Q(6, -5).

PQ  =  √[(x2 - x1)2 + (y2 - y1)2]

Step 2 :

Substitute x1 = -2, y1 = -3, x2 = 6, and y2 = -5.

PQ  =  √[(6 + 2)2 + (-5 + 3)2]

PQ  =  √(82 + (-2)2]

PQ  =  √(64 + 4)

PQ  =  √68

PQ  =  √(2 x 2 x 17)

PQ  =  2√17 units Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

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