**Pythagorean theorem in the coordinate plane :**

In this section, we are going to see, how Pythagorean theorem can be used to find lengths in a 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 : **

The figure shows a right triangle. Find the approximate length of the hypotenuse to the nearest tenth. Check your answer for reasonableness.

**Solution :**

**Step 1 : **

Find the length of each leg.

The length of the vertical leg is 4 units.

The length of the horizontal leg is 2 units.

**Step 2 :**

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 3 :**

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 4 :**

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

4.5 ≈ c

**Step 5 :**

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.

The hypotenuse is about 4.5 units long.

**Problem 2 : **

The figure shows a right triangle. Find the approximate length of the hypotenuse to the nearest tenth. Check your answer for reasonableness.

**Solution :**

**Step 1 :**

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 2 :**

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 3 :**

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 4 :**

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

6.4 ≈ c

**Step 5 :**

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.

The hypotenuse is about 6.4 units long.

After having gone through the stuff given above, we hope that the students would have understood "Pythagorean theorem in the coordinate plane".

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