1. The problem is to find the missing side lengths of four right triangles using the Pythagorean theorem. 2. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$): $$c^2 = a^2 + b^2$$ If the hypotenuse and one side are known, the other side can be found by: $$b^2 = c^2 - a^2$$ 3. For triangle a (sides 3 cm and 4 cm, missing side $c$): $$c^2 = 3^2 + 4^2 = 9 + 16 = 25$$ $$c = \sqrt{25} = 5$$ So, the missing side $c$ is 5 cm. 4. For triangle b (hypotenuse 6 cm and one side 6 cm, missing side $b$): $$b^2 = 6^2 - 6^2 = 36 - 36 = 0$$ $$b = \sqrt{0} = 0$$ So, the missing side $b$ is 0 cm, meaning the triangle is degenerate (the two sides are equal to the hypotenuse). 5. For triangle c (sides 7 m and 5 m, missing side $c$): Assuming 7 m is the hypotenuse (largest side), then: $$c^2 = 7^2 - 5^2 = 49 - 25 = 24$$ $$c = \sqrt{24} = 2\sqrt{6} \approx 4.90$$ So, the missing side $c$ is approximately 4.90 m. 6. For triangle d (sides 10 mm and 15 mm, missing side $d$): Assuming 15 mm is the hypotenuse: $$d^2 = 15^2 - 10^2 = 225 - 100 = 125$$ $$d = \sqrt{125} = 5\sqrt{5} \approx 11.18$$ So, the missing side $d$ is approximately 11.18 mm. Final answers: - a: $c = 5$ cm - b: $b = 0$ cm - c: $c \approx 4.90$ m - d: $d \approx 11.18$ mm