Question

Find the area of the shaded region to the nearest square unit. The base and height of the smaller shaded right triangle are the same length. There are 2 triangles connected at their base. A larger triangle has 2 side lengths of 14 feet. The smaller triangle has a side length of 7 feet, and the other 2 sides are equal. What is the area of the shaded region?

1) 123 sq. ft
2) 61 sq. ft
3) 49 sq. ft
4) 110 sq. ft

Answer 1

The area of the shaded region can be found by subtracting the area of the smaller triangle from the area of the larger triangle. Using the formula for the area of a triangle, the areas are calculated and subtracted to find the area of the shaded region. The answer is approximately 77 sq. ft.

Step-by-step explanation:

The area of the shaded region can be found by subtracting the area of the smaller triangle from the area of the larger triangle.

Using the formula for the area of a triangle (1/2 × base × height), the area of the larger triangle is 1/2 × 14 ft × 14 ft = 98 sq. ft.

The smaller triangle has a base of 7 ft and two congruent sides. We can use the Pythagorean Theorem to find the height:
h^2 = (7 ft)^2 - (7 ft/2)^2
h^2 = 49 ft^2 - 12.25 ft^2
h^2 = 36.75 ft^2
h ≈ 6.06 ft

Therefore, the area of the smaller triangle is 1/2 × 7 ft × 6.06 ft ≈ 21.21 sq. ft.

Finally, the area of the shaded region is 98 sq. ft - 21.21 sq. ft ≈ 76.79 sq. ft. To the nearest square unit, the area of the shaded region is approximately 77 sq. ft.