Pythagorean Inequality Theorem Worksheet with seven triangles to analyze.

This is a worksheet on the Pythagorean Inequality Theorem, which helps classify triangles as acute, obtuse, or right based on their side lengths.

The theorem states:
- If $a^2 + b^2 > c^2$, the triangle is acute.
- If $a^2 + b^2 = c^2$, the triangle is right.
- If $a^2 + b^2 < c^2$, the triangle is obtuse.

Here, $c$ is always the length of the longest side (the hypotenuse in a right triangle).

I will solve this problem by calculating the required values for each triangle and classifying it accordingly.

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Triangle 1: Sides 14 km, 8 km, 15 km

- Longest side $c = 15$ km
- Let $a = 14$ km, $b = 8$ km
- $a^2 = 196$
- $b^2 = 64$
- $a^2 + b^2 = 196 + 64 = 260$
- $c^2 = 225$
- Since $260 > 225$, the triangle is acute.

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Triangle 2: Sides 10 cm, 9 cm, 5 cm

- Longest side $c = 10$ cm
- Let $a = 9$ cm, $b = 5$ cm
- $a^2 = 81$
- $b^2 = 25$
- $a^2 + b^2 = 81 + 25 = 106$
- $c^2 = 100$
- Since $106 > 100$, the triangle is acute.

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Triangle 3: Sides 17 mi, 10 mi, 9 mi

- Longest side $c = 17$ mi
- Let $a = 10$ mi, $b = 9$ mi
- $a^2 = 100$
- $b^2 = 81$
- $a^2 + b^2 = 100 + 81 = 181$
- $c^2 = 289$
- Since $181 < 289$, the triangle is obtuse.

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Triangle 4: Sides 15 yd, 13 yd, 17 yd

- Longest side $c = 17$ yd
- Let $a = 15$ yd, $b = 13$ yd
- $a^2 = 225$
- $b^2 = 169$
- $a^2 + b^2 = 225 + 169 = 394$
- $c^2 = 289$
- Since $394 > 289$, the triangle is acute.

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Triangle 5: Sides 9 mi, 7 mi, 12 mi

- Longest side $c = 12$ mi
- Let $a = 9$ mi, $b = 7$ mi
- $a^2 = 81$
- $b^2 = 49$
- $a^2 + b^2 = 81 + 49 = 130$
- $c^2 = 144$
- Since $130 < 144$, the triangle is obtuse.

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Triangle 6: Sides 13 cm, 13 cm, 12 cm

- Longest side $c = 13$ cm (either of the two equal sides)
- Let $a = 13$ cm, $b = 12$ cm
- $a^2 = 169$
- $b^2 = 144$
- $a^2 + b^2 = 169 + 144 = 313$
- $c^2 = 169$
- Since $313 > 169$, the triangle is acute.

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Triangle 7: Sides 6, 8, 10

- Longest side $c = 10$
- Let $a = 6$, $b = 8$
- $a^2 = 36$
- $b^2 = 64$
- $a^2 + b^2 = 36 + 64 = 100$
- $c^2 = 100$
- Since $100 = 100$, the triangle is right.

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Final Answer:

Here is the completed chart:

| Triangle | $a^2$ | $b^2$ | $a^2 + b^2$ | $a^2 + b^2 \ ? \ c^2$ | $c^2$ | Acute, Obtuse, Right Triangle? |
|----------|--------|--------|--------------|--------------------------|--------|--------------------------------|
| 1. | 196 | 64 | 260 | > | 225 | Acute |
| 2. | 81 | 25 | 106 | > | 100 | Acute |
| 3. | 100 | 81 | 181 | < | 289 | Obtuse |
| 4. | 225 | 169 | 394 | > | 289 | Acute |
| 5. | 81 | 49 | 130 | < | 144 | Obtuse |
| 6. | 169 | 144 | 313 | > | 169 | Acute |
| 7. | 36 | 64 | 100 | = | 100 | Right |

This completes the worksheet.