The Converse of Pythagoras Theorem - Variation Theory - Free Printable

To determine whether each triangle is right-angled, I will apply the Pythagorean theorem. This theorem states that for a triangle to be right-angled, the square of the longest side (the hypotenuse) must equal the sum of the squares of the other two sides.

The formula is:
$$
a^2 + b^2 = c^2
$$
where $c$ is the hypotenuse (longest side), and $a$ and $b$ are the other two sides.

I will evaluate each triangle one by one.

---

Triangle 1: Sides 8 cm, 15 cm, 17 cm

- Longest side: 17 cm
- Check: $8^2 + 15^2 = 64 + 225 = 289$
- $17^2 = 289$
- ✔ Right-angled

---

Triangle 2: Sides 10 cm, 15 cm, 18 cm

- Longest side: 18 cm
- Check: $10^2 + 15^2 = 100 + 225 = 325$
- $18^2 = 324$
- ✘ Not equal → Not right-angled

---

Triangle 3: Sides 39 cm, 80 cm, 89 cm

- Longest side: 89 cm
- Check: $39^2 + 80^2 = 1521 + 6400 = 7921$
- $89^2 = 7921$
- ✔ Right-angled

---

Triangle 4: Sides 3 cm, 7 cm, 10 cm

- Longest side: 10 cm
- Check: $3^2 + 7^2 = 9 + 49 = 58$
- $10^2 = 100$
- ✘ Not equal → Not right-angled

---

Triangle 5: Sides 10 cm, 10 cm, 200 cm

- Longest side: 200 cm
- Check: $10^2 + 10^2 = 100 + 100 = 200$
- $200^2 = 40,000$
- ✘ Not equal → Not right-angled

Also, note: this triangle violates the triangle inequality (sum of two sides must be greater than the third side). $10 + 10 = 20 < 200$, so it’s not even a valid triangle.

---

Triangle 6: Sides 35 cm, 84 cm, 91 cm

- Longest side: 91 cm
- Check: $35^2 + 84^2 = 1225 + 7056 = 8281$
- $91^2 = 8281$
- ✔ Right-angled

---

Triangle 7: Sides 65 cm, 72 cm, 97 cm

- Longest side: 97 cm
- Check: $65^2 + 72^2 = 4225 + 5184 = 9409$
- $97^2 = 9409$
- ✔ Right-angled

---

Triangle 8: Sides 36 cm, 77 cm, 85 cm

- Longest side: 85 cm
- Check: $36^2 + 77^2 = 1296 + 5929 = 7225$
- $85^2 = 7225$
- ✔ Right-angled

---

Triangle 9: Sides 40 cm, 50 cm, 0.03 m

First, convert all units to the same measure.
- 0.03 m = 3 cm
- Sides: 3 cm, 40 cm, 50 cm
- Longest side: 50 cm
- Check: $3^2 + 40^2 = 9 + 1600 = 1609$
- $50^2 = 2500$
- ✘ Not equal → Not right-angled

---

Triangle 10: Sides 5 cm, 12 cm, 60 cm

- Longest side: 60 cm
- Check: $5^2 + 12^2 = 25 + 144 = 169$
- $60^2 = 3600$
- ✘ Not equal → Not right-angled

Also, violates triangle inequality: $5 + 12 = 17 < 60$

---

Triangle 11: Sides 0.12 m, 0.37 m, 35 cm

Convert all to meters or centimeters. Let’s use centimeters:
- 0.12 m = 12 cm
- 0.37 m = 37 cm
- 35 cm remains
- Sides: 12 cm, 35 cm, 37 cm
- Longest side: 37 cm
- Check: $12^2 + 35^2 = 144 + 1225 = 1369$
- $37^2 = 1369$
- ✔ Right-angled

---

## Final Summary:

✔ Right-angled triangles: 1, 3, 6, 7, 8, 11

✘ Not right-angled: 2, 4, 5, 9, 10

*(Note: Triangles 5 and 10 are also invalid due to violating triangle inequality.)*