Pythagoras Theorem Squaring a Number and Square Roots - ppt download - Free Printable

Let's solve each of the four examples using Pythagoras' Theorem, which states:

> In a right-angled triangle:
> $$
> a^2 + b^2 = c^2
> $$
> where:
> - $ c $ is the longest side (hypotenuse),
> - $ a $ and $ b $ are the shorter sides.

We'll follow the steps as indicated in the image:
- For longest side: Square → Add → Square Root
- For shorter side: Square → Subtract → Square Root

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✔ Example 1

Triangle with legs: 7 cm and 9 cm. Find hypotenuse $ x $.

This is finding the longest side, so use:
- Step 1: Square the two shorter sides
- Step 2: Add them
- Step 3: Take square root

Step 1:
$ 7^2 = 49 $
$ 9^2 = 81 $

Step 2:
$ 49 + 81 = 130 $

Step 3:
$ \sqrt{130} \approx 11.40 $ cm

✔ So, $ x = \sqrt{130} \approx 11.40 $ cm

Answer:
1. $ 7^2 = 49 $, $ 9^2 = 81 $
2. $ 49 + 81 = 130 $
3. $ x = \sqrt{130} \approx 11.40 $ cm

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✔ Example 2

Triangle with legs: 4 cm and 8 cm. Find hypotenuse $ x $.

Again, longest side unknown.

Step 1:
$ 4^2 = 16 $
$ 8^2 = 64 $

Step 2:
$ 16 + 64 = 80 $

Step 3:
$ \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} \approx 8.94 $ cm

✔ So, $ x = \sqrt{80} \approx 8.94 $ cm

Answer:
1. $ 4^2 = 16 $, $ 8^2 = 64 $
2. $ 16 + 64 = 80 $
3. $ x = \sqrt{80} \approx 8.94 $ cm

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✔ Example 3

Triangle with one leg = 7 cm, hypotenuse = 12 cm. Find the other leg $ x $.

Here, we’re finding a shorter side, so:
- Step 1: Square both known sides
- Step 2: Subtract (larger minus smaller)
- Step 3: Square root

Step 1:
$ 12^2 = 144 $
$ 7^2 = 49 $

Step 2:
$ 144 - 49 = 95 $

Step 3:
$ \sqrt{95} \approx 9.75 $ cm

✔ So, $ x = \sqrt{95} \approx 9.75 $ cm

Answer:
1. $ 12^2 = 144 $, $ 7^2 = 49 $
2. $ 144 - 49 = 95 $
3. $ x = \sqrt{95} \approx 9.75 $ cm

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✔ Example 4

Triangle with one leg = 15 mm, hypotenuse = 23 mm. Find the other leg $ x $.

Again, finding a shorter side.

Step 1:
$ 23^2 = 529 $
$ 15^2 = 225 $

Step 2:
$ 529 - 225 = 304 $

Step 3:
$ \sqrt{304} \approx 17.44 $ mm

✔ So, $ x = \sqrt{304} \approx 17.44 $ mm

Answer:
1. $ 23^2 = 529 $, $ 15^2 = 225 $
2. $ 529 - 225 = 304 $
3. $ x = \sqrt{304} \approx 17.44 $ mm

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✔ Final Answers Summary:

#### Example 1:
1. $ 7^2 = 49 $, $ 9^2 = 81 $
2. $ 49 + 81 = 130 $
3. $ x = \sqrt{130} \approx 11.40 $ cm

#### Example 2:
1. $ 4^2 = 16 $, $ 8^2 = 64 $
2. $ 16 + 64 = 80 $
3. $ x = \sqrt{80} \approx 8.94 $ cm

#### Example 3:
1. $ 12^2 = 144 $, $ 7^2 = 49 $
2. $ 144 - 49 = 95 $
3. $ x = \sqrt{95} \approx 9.75 $ cm

#### Example 4:
1. $ 23^2 = 529 $, $ 15^2 = 225 $
2. $ 529 - 225 = 304 $
3. $ x = \sqrt{304} \approx 17.44 $ mm

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