A triangle and a parallelogram have the same base and the same area. If the sides of the triangles are 26 cm, 28 cm and 30 cm and the parallelogram stands on the base 28 cm, then the height of the parallelogram is:

Question

Accepted Answer

To determine the height of the parallelogram, we need to first calculate the area of the triangle, as the triangle and the parallelogram have the same area.

The sides of the triangle are given as 26 cm, 28 cm, and 30 cm. We can use Heron's formula to find the area of the triangle. Heron's formula states that the area of a triangle with sides $a$, $b$, and $c$ is:

$$ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} $$

where $s$ is the semi-perimeter of the triangle:

$$ s = \frac{a + b + c}{2} = \frac{26 + 28 + 30}{2} = 42 \, \text{cm} $$

Now, we can calculate the area:

$$ \text{Area} = \sqrt{42(42-26)(42-28)(42-30)} $$
$$ = \sqrt{42 \times 16 \times 14 \times 12} $$
$$ = \sqrt{112896} $$
$$ = 336 \, \text{cm}^2 $$

The area of the triangle is 336 cm². Since the parallelogram has the same area and stands on a base of 28 cm, we can use the formula for the area of a parallelogram, which is:

$$ \text{Area} = \text{base} \times \text{height} $$

Given that the area is 336 cm² and the base is 28 cm, we can solve for the height:

$$ 336 = 28 \times \text{height} $$
$$ \text{height} = \frac{336}{28} $$
$$ \text{height} = 12 \, \text{cm} $$

Therefore, the height of the parallelogram is 12 cm, which corresponds to Option 4.

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

1. **Option 1 (10 cm):** If the height were 10 cm, the area of the parallelogram would be $28 \times 10 = 280 \, \text{cm}^2$, which does not match the area of the triangle (336 cm²).

2. **Option 2 (8 cm):** If the height were 8 cm, the area of the parallelogram would be $28 \times 8 = 224 \, \text{cm}^2$, which is also incorrect as it does not match the area of the triangle.

3. **Option 3 (7 cm):** If the height were 7 cm, the area of the parallelogram would be $28 \times 7 = 196 \, \text{cm}^2$, which is again incorrect as it does not match the area of the triangle.

Thus, Option 4 is the only correct answer, as it provides the height that results in the parallelogram having the same area as the triangle.