1. **Problem statement:** Find the value of $k$ and the area of a trapezium given certain conditions (not specified in the question, so we assume $k$ is a parameter related to the trapezium's dimensions). 2. **Formula for trapezium area:** The area $A$ of a trapezium with parallel sides $a$ and $b$, and height $h$ is given by: $$A = \frac{1}{2} (a + b) h$$ 3. **Finding $k$:** Usually, $k$ might represent a length or a ratio in the trapezium. Without specific details, we cannot find $k$ explicitly. Please provide the trapezium's side lengths or coordinates. 4. **Example:** Suppose the trapezium has parallel sides $a = k$, $b = 10$, and height $h = 5$. Then the area is: $$A = \frac{1}{2} (k + 10) \times 5 = \frac{5}{2} (k + 10)$$ 5. **If the area is known, say $A = 60$, solve for $k$:** $$60 = \frac{5}{2} (k + 10)$$ $$60 = \frac{5}{2}k + 25$$ $$60 - 25 = \frac{5}{2}k$$ $$35 = \frac{5}{2}k$$ $$k = 35 \times \frac{2}{5} = 14$$ 6. **Final answer:** $k = 14$ and the trapezium area is $60$ (given the example values). Please provide the exact values or conditions to find $k$ and the trapezium area precisely.