Question

the triangle inscribed within square abcd has, as its base, side ad, and has a height of 6 cm. if the area of the triangle is 21 cm\\(^2\\), what is the area of the shaded region?

a. 28 cm\\(^2\\)
b. 22 cm\\(^2\\)
c. 18 cm\\(^2\\)
d. 21 cm\\(^2\\)

Explanation:

Find the base of the triangle

The area of a triangle is given by:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$

Given that the area is \(21\text{ cm}^2\) and the height is \(6\text{ cm}\):

$$ 21 = \frac{1}{2} \times AD \times 6 $$

$$ 21 = 3 \times AD $$

$$ AD = 7\text{ cm} $$

Calculate the area of the square

Since \(ABCD\) is a square, all its sides are equal to the base \(AD\):

$$ \text{Side length} = 7\text{ cm} $$

The area of the square is:

$$ \text{Area}_{\text{square}} = \text{side}^2 = 7^2 = 49\text{ cm}^2 $$

Calculate the area of the shaded region

The shaded region is the area of the square minus the area of the triangle:

$$ \text{Area}_{\text{shaded}} = \text{Area}_{\text{square}} - \text{Area}_{\text{triangle}} $$

$$ \text{Area}_{\text{shaded}} = 49 - 21 = 28\text{ cm}^2 $$

Answer:

• (A) \(28\text{ cm}^2\) (Correct answer)
• (B) \(22\text{ cm}^2\)
• (C) \(18\text{ cm}^2\)
• (D) \(21\text{ cm}^2\)