The area of a trapezium is $1586 cm^2$ and the perpendicular distance between its parallel sides is 26 cm. If one of the parallel sides is 84 cm, the other side is equal to :

Question

Accepted Answer

To determine the length of the other parallel side of the trapezium, we can use the formula for the area of a trapezium:

$$ \text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height} $$

Given:
- Area = $1586 \, \text{cm}^2$
- Height (perpendicular distance between the parallel sides) = $26 \, \text{cm}$
- One of the parallel sides ($\text{Base}_1$) = $84 \, \text{cm}$

We need to find the length of the other parallel side ($\text{Base}_2$).

Substitute the known values into the formula:

$$ 1586 = \frac{1}{2} \times (84 + \text{Base}_2) \times 26 $$

First, simplify the equation:

$$ 1586 = 13 \times (84 + \text{Base}_2) $$

Divide both sides by 13 to isolate the sum of the bases:

$$ \frac{1586}{13} = 84 + \text{Base}_2 $$

$$ 122 = 84 + \text{Base}_2 $$

Now, solve for $\text{Base}_2$:

$$ \text{Base}_2 = 122 - 84 $$

$$ \text{Base}_2 = 38 \, \text{cm} $$

Therefore, the length of the other parallel side is $38 \, \text{cm}$, which corresponds to Option 3.

**Explanation for why other options are incorrect:**

1. **Option 1 (74 cm):** If $\text{Base}_2$ were 74 cm, the sum of the bases would be $84 + 74 = 158$. Substituting back into the area formula, the area would be $\frac{1}{2} \times 158 \times 26 = 2054 \, \text{cm}^2$, which does not match the given area of $1586 \, \text{cm}^2$.

2. **Option 2 (76 cm):** If $\text{Base}_2$ were 76 cm, the sum of the bases would be $84 + 76 = 160$. Substituting back into the area formula, the area would be $\frac{1}{2} \times 160 \times 26 = 2080 \, \text{cm}^2$, which is incorrect.

3. **Option 4 (37 cm):** If $\text{Base}_2$ were 37 cm, the sum of the bases would be $84 + 37 = 121$. Substituting back into the area formula, the area would be $\frac{1}{2} \times 121 \times 26 = 1573 \, \text{cm}^2$, which is also incorrect.

Thus, Option 3 (38 cm) is the only correct answer that satisfies the given conditions.