In quadrilateral ABCD, the diagonal BD measures $70 ext{ cm}$. The perpendiculars from vertices A and C to diagonal BD are $AL = 18 ext{ cm}$ and $CM = 20 ext{ cm}$ respectively. Calculate the area of quadrilateral ABCD using the formula:

$$ ext{Area} = \frac{1}{2} imes ext{Diagonal} imes ( ext{Sum of Perpendiculars})$$

Question

In quadrilateral ABCD, the diagonal BD measures $70 	ext{ cm}$. The perpendiculars from vertices A and C to diagonal BD are $AL = 18 	ext{ cm}$ and $CM = 20 	ext{ cm}$ respectively. Calculate the area of quadrilateral ABCD using the formula:

$$	ext{Area} = \frac{1}{2} 	imes 	ext{Diagonal} 	imes (	ext{Sum of Perpendiculars})$$

Accepted Answer

**Solution:**

1. Given:
   - Diagonal $BD = 70 	ext{ cm}$
   - Perpendicular from $A$ to $BD$, $AL = 18 	ext{ cm}$
   - Perpendicular from $C$ to $BD$, $CM = 20 	ext{ cm}$

2. The formula for the area of quadrilateral $ABCD$ when diagonal $BD$ and perpendiculars from $A$ and $C$ to $BD$ are known is:

$$	ext{Area} = \frac{1}{2} 	imes BD 	imes (AL + CM)$$

3. Substitute the values:

$$	ext{Area} = \frac{1}{2} 	imes 70 	imes (18 + 20) = \frac{1}{2} 	imes 70 	imes 38$$

4. Calculate:

$$	ext{Area} = 35 	imes 38 = 1330 	ext{ cm}^2$$

**Answer:** $1330 	ext{ cm}^2$

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