In the diagram, ABC is an equilateral triangle. Find the value of (x+y).

Here are the solutions to the problems: Problem 1: a) Given that P = \factors of 36\ and Q = \factors of 54\. i) List the members in the sets P and Q. Step 1: List the factors of 36. P = \1, 2, 3, 4, 6, 9, 12, 18, 36\ Step 2: List the factors of 54. Q = \1, 2, 3, 6, 9, 18, 27, 54\ The members are: P = \1, 2, 3, 4, 6, 9, 12, 18, 36\ Q = \1, 2, 3, 6, 9, 18, 27, 54\ ii) Find: () P Q; Step 1: Identify the common factors in sets P and Q. P Q = \1, 2, 3, 6, 9, 18\ P Q = \1, 2, 3, 6, 9, 18\ () n(P Q); Step 1: Count the number of elements in P Q. There are 6 elements in P Q. n(P Q) = 6 () the Highest Common Factor (HCF) of 36 and 54. Step 1: The HCF is the largest number in the set P Q. The largest common factor is 18. HCF of 36 and 54 is 18. b) Write down the next two terms of the sequence 1, 4, 9, ... Step 1: Identify the pattern of the sequence. The sequence consists of perfect squares: 1^2=1, 2^2=4, 3^2=9. Step 2: Calculate the next two terms. The next terms will be 4^2 and 5^2. 4^2 = 16 5^2 = 25 The next two terms are 16, 25. c) The median of the ordered set of observations 2, 3, (4m - 3), (3m + 1), 11 and 13 in ascending order is 6. Find the value of m. Step 1: Identify the middle terms. There are 6 observations, so the median is the average of the 3rd and 4th terms. The 3rd term is (4m - 3) and the 4th term is (3m + 1). Step 2: Set up the equation for the median. Median = ((4m - 3) + (3m + 1))/(2) Given that the median is 6: 6 = (4m - 3 + 3m + 1)/(2) Step 3: Solve for m. 6 = (7m - 2)/(2) 12 = 7m - 2 12 + 2 = 7m 14 = 7m m = (14)/(7) m = 2 The value of m is 2. --- Problem 2: a) Simplify: ((1)/(3) + (1)/(12)) ÷ ((2)/(3) - (5)/(8)). Step 1: Simplify the first parenthesis. (1)/(3) + (1)/(12) = (4)/(12) + (1)/(12) = (5)/(12) Step 2: Simplify the second parenthesis. The least common multiple of 3 and 8 is 24. (2)/(3) - (5)/(8) = (2 × 8)/(24) - (5 × 3)/(24) = (16)/(24) - (15)/(24) = (1)/(24) Step 3: Perform the division. (5)/(12) ÷ (1)/(24) = (5)/(12) × (24)/(1) = 5 × 2 = 10 The simplified expression is 10. b) Find the product of (2x - 3) and (2x + 3). Step 1: Recognize the pattern as a difference of squares, (a-b)(a+b) = a^2 - b^2. Here, a = 2x and b = 3. Step 2: Apply the formula. (2x - 3)(2x + 3) = (2x)^2 - 3^2 = 4x^2 - 9 The product is 4x^2 - 9. c) In the diagram, ABC is an equilateral triangle. Find the value of (x+y). The sides of the triangle are AB = (3x+2) cm, AC = 11 cm, and BC = (2y+3) cm. Step 1: Use the property of an equilateral triangle that all sides are equal. 3x + 2 = 11 2y + 3 = 11 Step 2: Solve the equation for x. 3x + 2 = 11 3x = 11 - 2 3x = 9 x = (9)/(3) x = 3 Step 3: Solve the equation for y. 2y + 3 = 11 2y = 11 - 3 2y = 8 y = (8)/(2) y = 4 Step 4: Calculate (x+y). x+y = 3+4 = 7 The value of (x+y) is 7. --- Problem 3: a) Given the relation L = (2(m^2 - n^2))/(4(m + n)). i) Simplify L. Step 1 ✂️ _That answer was long and got cut off. Reply continue and I'll finish it._