1. **Problem 1: Calculate the value of the pronumeral $x$ using Pythagoras' Theorem.** 2. Pythagoras' Theorem states that in a right triangle, the square of the hypotenuse ($c$) equals the sum of the squares of the other two sides ($a$ and $b$): $$c^2 = a^2 + b^2$$ 3. Here, the hypotenuse is 90 cm, one side is 72 cm, and the other side is $x$ cm. So: $$90^2 = 72^2 + x^2$$ 4. Calculate the squares: $$8100 = 5184 + x^2$$ 5. Subtract 5184 from both sides: $$8100 - 5184 = x^2$$ $$2916 = x^2$$ 6. Take the square root of both sides: $$x = \sqrt{2916}$$ 7. Calculate the square root: $$x = 54$$ --- 1. **Problem 2: Find the perimeter of the composite shape to the nearest whole number.** 2. The composite shape consists of a rectangle and a right triangle joined together. 3. The rectangle has a height of 8 cm, and the triangle has a base of 10 cm. 4. To find the perimeter, add all the outer sides: - Rectangle vertical side: 8 cm - Rectangle horizontal side (unknown, but assume the base of the rectangle is the same as the triangle's height, which we need to find) - Triangle base: 10 cm - Triangle hypotenuse (use Pythagoras to find this): Let the triangle height be $h$ (same as rectangle base), then: $$hypotenuse = \sqrt{10^2 + h^2}$$ 5. Since the rectangle height is 8 cm, the triangle height is also 8 cm. 6. Calculate the hypotenuse: $$hypotenuse = \sqrt{10^2 + 8^2} = \sqrt{100 + 64} = \sqrt{164} \approx 12.81$$ 7. Now add all sides for the perimeter: $$Perimeter = 8 + 10 + 8 + 12.81 = 38.81$$ 8. Round to the nearest whole number: $$Perimeter \approx 39$$