Mensuration - Area of Parallelograms and Triangles

A parallelogram has a base of $15\text{ cm}$ and a corresponding height of $8\text{ cm}$. Find its area. If another side of the parallelogram is $10\text{ cm}$, find the height corresponding to that side.

1. Find the area using the first base and height: $Area = Base \times Height = 15 \times 8 = 120\text{ cm}^2$
2. Use the area to find the second height for the side $10\text{ cm}$: $Area = Base_2 \times Height_2$ $120 = 10 \times Height_2$ $Height_2 = \frac{120}{10} = 12\text{ cm}$

First, calculate the total area using the known pair of base and height. Since the area of the parallelogram remains constant regardless of which side is used as the base, use that area value to solve for the missing height corresponding to the second side.

The area of a triangle is $50\text{ cm}^2$. If the base of the triangle is $12.5\text{ cm}$, calculate its altitude (height).

1. Write down the area formula for a triangle: $Area = \frac{1}{2} \times Base \times Height$
2. Substitute the given values: $50 = \frac{1}{2} \times 12.5 \times Height$
3. Solve for Height: $100 = 12.5 \times Height$ $Height = \frac{100}{12.5} = 8\text{ cm}$

Plug the known area and base into the triangle area formula. Multiply the area by $2$ to remove the fraction, then divide by the base to find the perpendicular height.

$Area = \frac{1}{2} \times Base \times Height$ $Area = \frac{1}{2} \times 14 \times 9$ $Area = 7 \times 9$ $Area = 63\text{ cm}^2$

We use the standard triangle area formula. Half of the base ($14\text{ cm}$) is $7\text{ cm}$, which when multiplied by the height ($9\text{ cm}$) gives $63\text{ cm}^2$.

$Base = \frac{Area}{Height}$ $Base = \frac{120}{8}$ $Base = 15\text{ cm}$

To find the base of a parallelogram when area and height are given, divide the area by the height.