Problems involving similar triangles and prism volume calculations.

Here are the solutions to the problems. Problem 7: Step 1: Identify similar triangles. Since AB is parallel to CD, triangles ATB and DTC are similar. This means that the ratio of corresponding sides is equal. (AT)/(TD) = (BT)/(TC) Step 2: Use the given ratio and total length. We are given AT : TD = 1 : 3, so (AT)/(TD) = (1)/(3). Therefore, (BT)/(TC) = (1)/(3). This implies TC = 3 × BT. We are also given that CB = 12 cm. We know that CB = BT + TC. Substitute TC = 3 × BT into the equation: BT + 3 × BT = 12 4 × BT = 12 Step 3: Calculate the length of TB. BT = (12)/(4) BT = 3 cm The length of TB is 3 cm. Problem 8: Step 1: Calculate the area of the equilateral triangle cross-section. The formula for the area of an equilateral triangle with side s is A = sqrt(3)4s^2. Given side s = 3 cm. A = sqrt(3)4(3)^2 A = sqrt(3)4(9) A = 9sqrt(3)4 cm^2 Step 2: Calculate the volume of the prism. The volume of a prism is given by the formula V = Area of cross-section × Length. Given length = 25 cm. V = 9sqrt(3)4 × 25 V = 225sqrt(3)4 cm^3 As a decimal, V ≈ (225 × 1.73205)/(4) ≈ (389.71125)/(4) ≈ 97.4278 cm^3. The volume of the prism is 225sqrt(3)4 cm^3 or approximately 97.43 cm^3. Problem 9: Step 1: Find the prime factorization of each capacity. We need to find the greatest common divisor (GCD) of 36, 84, and 90. 36 = 2 × 18 = 2 × 2 × 9 = 2^2 × 3^2 84 = 2 × 42 = 2 × 2 × 21 = 2^2 × 3 × 7 90 = 2 × 45 = 2 × 3 × 15 = 2 × 3 × 3 × 5 = 2 × 3^2 × 5 Step 2: Identify the common prime factors and their lowest powers. The common prime factors are 2 and 3. The lowest power of 2 present in all factorizations is 2^1. The lowest power of 3 present in all factorizations is 3^1. Step 3: Multiply the common prime factors with their lowest powers. GCD(36, 84, 90) = 2^1 × 3^1 = 2 × 3 = 6 The capacity of the greatest vessel is 6 litres. That's 2 down. 3 left today — send the next one.