**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

**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

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.

**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 : **

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

**AB = **√[(2-(-12)]² + (5-3)²

AB = √[(2+12)]² + (2)²

AB = √[(14)]² + 4

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 : **

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

**PQ = **√[(6-(-2)]² + [(-5-(-3)]²

PQ = √(6+2)² + (-5+3)²

PQ = √(8)² + (-5+3)²

PQ = √(8)² + (-2)²

PQ = √64 + 4

PQ = √68

PQ = √(2 x 2 x 17)

PQ = 2√17 units

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