**Using Pythagorean theorem to find 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.

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

If a and b are legs and c is the hypotenuse, then

a² + b² = c²

**Problem 1 : **

Find the distance between the points (1, 3) and (-1, -1) using Pythagorean theorem. Check your answer for reasonableness.

**Solution : **

**Step 1 : **

Locate the points (1, 3) and (-1, -1) on a coordinate plane.

**Step 2 :**

Draw horizontal segment of length 2 units from (-1, -1) and vertical segment of length of 4 units from (1, 3) as shown in the figure.

**Step 3 :**

In step 2, a right triangle is formed.

The length of the vertical leg is 4 units.

The length of the horizontal leg is 5 units.

**Step 4 :**

Let a = 4 and b = 2 and c represent the length of the hypotenuse.

Because a and b are legs and c is hypotenuse, by Pythagorean Theorem, we have

a² + b² = c²

**Step 5 :**

Plug a = 4 and b = 2 in a² + b² = c² to solve for c.

4² + 2² = c²

Simplify.

16 + 4 = c²

20 = c²

Take the square root of both sides.

√20 = √c²

√20 = c

**Step 6 :**

Find the value of √20 using calculator and round to the nearest tenth

4.5 ≈ c

**Step 7 :**

Check for reasonableness by finding perfect squares close to 20.

√20 is between √16 and √25, so 4 < √20 < 5.

Since 4.5 is between 4 and 5, the answer is reasonable.

Hence, the distance between the points (1, 3) and (-1, -1) is about 4.5 units.

**Problem 2 : **

Find the distance between the points (-3, 2) and (2, -2) using Pythagorean theorem. Check your answer for reasonableness.

**Solution : **

**Step 1 :**

Locate the points (-3, 2) and (2, -2) on a coordinate plane.

**Step 2 :**

Draw horizontal segment of length 5 units from (-3, -2) and vertical segment of length of 4 units from (2, -2) as shown in the figure.

**Step 3 :**

Find the length of each leg.

The length of the vertical leg is 4 units.

The length of the horizontal leg is 5 units.

**Step 4 :**

Let a = 4 and b = 5 and c represent the length of the hypotenuse.

Because a and b are legs and c is hypotenuse, by Pythagorean Theorem, we have

a² + b² = c²

**Step 5 :**

Plug a = 4 and b = 5 in a² + b² = c² to solve for c.

4² + 5² = c²

Simplify.

16 + 25 = c²

41 = c²

Take the square root of both sides.

√41 = √c²

√41 = c

**Step 6 :**

Find the value of √41 using calculator and round to the nearest tenth

6.4 ≈ c

**Step 7 :**

Check for reasonableness by finding perfect squares close to 41.

√41 is between √36 and √49, so 6 < √41 < 7.

Since 6.4 is between 6 and 7, the answer is reasonable.

Hence, the distance between the points (-3, 2) and (2, -2) is about 4.5 units.

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