Two rhombuses PQRS and MLKJ have sides calculated as follows:

- Side of rhombus PQRS is calculated using the formula $\sqrt{\frac{6}{2} + \frac{8}{2}} = \sqrt{10} = 5 ext{ cm}$.
- For rhombus MLKJ, given $\sin 30^\circ = \frac{ ext{perpendicular}}{ ext{hypotenuse}} = \frac{1}{2} = \frac{7.5}{ ext{side of MLKJ}}$, the side of MLKJ is found to be $15 ext{ cm}$.

If AB is the side of PQRS and CD is the side of MLKJ, what is the length of the median XY of trapezium ABCD formed by these sides?

![Screenshot 2025-09-03 01.54.55.png](https://res.cloudinary.com/aptidude/image/upload/v1756844761/aptidude_questions/lq6ijftoxflxvdjgxswg.png)

Calculate the median XY using the formula:

$$XY = \frac{AB + CD}{2}$$

Question

Two rhombuses PQRS and MLKJ have sides calculated as follows:

- Side of rhombus PQRS is calculated using the formula $\sqrt{\frac{6}{2} + \frac{8}{2}} = \sqrt{10} = 5 	ext{ cm}$.
- For rhombus MLKJ, given $\sin 30^\circ = \frac{	ext{perpendicular}}{	ext{hypotenuse}} = \frac{1}{2} = \frac{7.5}{	ext{side of MLKJ}}$, the side of MLKJ is found to be $15 	ext{ cm}$.

If AB is the side of PQRS and CD is the side of MLKJ, what is the length of the median XY of trapezium ABCD formed by these sides?

![Screenshot 2025-09-03 01.54.55.png](https://res.cloudinary.com/aptidude/image/upload/v1756844761/aptidude_questions/lq6ijftoxflxvdjgxswg.png)

Calculate the median XY using the formula:

$$XY = \frac{AB + CD}{2}$$

Accepted Answer

**Solution:**

1. Calculate the side of rhombus PQRS:
   - Given expression: $\sqrt{\frac{6}{2} + \frac{8}{2}} = \sqrt{3 + 4} = \sqrt{7}$ (Note: The original calculation shows $\sqrt{10}$, so assuming the sum is $10$)
   - As per the problem, side PQRS = $5 	ext{ cm}$.

2. Calculate the side of rhombus MLKJ using $\sin 30^\circ$:
   - $\sin 30^\circ = \frac{1}{2} = \frac{7.5}{	ext{side of MLKJ}}$
   - Side of MLKJ = $7.5 	imes 2 = 15 	ext{ cm}$.

3. Given AB = side of PQRS = $5 	ext{ cm}$ and CD = side of MLKJ = $15 	ext{ cm}$.

4. Median XY of trapezium ABCD is:
   $$XY = \frac{AB + CD}{2} = \frac{5 + 15}{2} = \frac{20}{2} = 10 	ext{ cm}$$

**Answer:** $10 	ext{ cm}$

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