1. **State the problem:** We need to determine which set of three side lengths can form a right-angled triangle. 2. **Recall the Pythagorean theorem:** For a triangle with sides $a$, $b$, and hypotenuse $c$, the triangle is right-angled if and only if $$a^2 + b^2 = c^2$$ where $c$ is the longest side. 3. **Check each option:** - a. Sides: 6.75 cm, 15 cm, 24 cm. Longest side is 24 cm. Calculate: $$6.75^2 + 15^2 = 45.5625 + 225 = 270.5625$$ Compare with $$24^2 = 576$$ Since $270.5625 \neq 576$, not right-angled. - b. Sides: 5.25 cm, 9 cm, 11.25 cm. Longest side is 11.25 cm. Calculate: $$5.25^2 + 9^2 = 27.5625 + 81 = 108.5625$$ Compare with $$11.25^2 = 126.5625$$ Since $108.5625 \neq 126.5625$, not right-angled. - c. Sides: 2.25 cm, 3 cm, 3.75 cm. Longest side is 3.75 cm. Calculate: $$2.25^2 + 3^2 = 5.0625 + 9 = 14.0625$$ Compare with $$3.75^2 = 14.0625$$ Since $14.0625 = 14.0625$, this set forms a right-angled triangle. - d. Sides: 3.75 cm, 5.25 cm, 6 cm. Longest side is 6 cm. Calculate: $$3.75^2 + 5.25^2 = 14.0625 + 27.5625 = 41.625$$ Compare with $$6^2 = 36$$ Since $41.625 \neq 36$, not right-angled. - e. Sides: 4.5 cm, 6 cm, 9 cm. Longest side is 9 cm. Calculate: $$4.5^2 + 6^2 = 20.25 + 36 = 56.25$$ Compare with $$9^2 = 81$$ Since $56.25 \neq 81$, not right-angled. 4. **Conclusion:** Only option c (2.25 cm, 3 cm, 3.75 cm) satisfies the Pythagorean theorem and can be used to construct a right-angled triangle.