Please help! Find the area and the perimeter of the shaded regions below. Give your answer as a completely simplified exact value in terms of π (no approximations). The figures below are based on semicircles or quarter circles and problems b), c), and d) are involving portions of a square. Please answer with your work, explanation, and answers to perimeter and area to the shaded region.

Part A

The perimeter of the of the shaded region is (3·π + 6 + 6·(√2)) cm

Part B

Step-by-step explanation:

Part A

The perimeter of the the shaded region is given as follows;

The perimeter of the the shaded region = The perimeter of the semicircle + The perimeter of the right triangle with legs 6 cm each - The length of the dotted line

The perimeter of the semicircle = π × r

We note that the diameter of the semicircle, D =   = 6 cm

The radius of the semicircle, r = D/2 = 6 cm/2 = 3 cm

∴ The perimeter of the semicircle = π × 3 cm = 3·π cm

The perimeter of the right triangle with legs 6 cm each = The length of the 2 6 cm legs + The length of the hypotenuse side

By Pythagoras' theorem, the length of the hypotenuse side = √(6² + 6²) = 6·(√2) cm

∴ The perimeter of the right triangle = 6 cm + 6 cm + 6·(√2) cm = (12 + 6·(√2)) cm

The length of the dotted line = The the length of side of the square from which the right triangle ABC is based

∴ The length of the dotted line = The length of = 6 cm

∴ The perimeter of the figure = 3·π cm + (12 + 6·(√2)) cm - 6 cm = (3·π + 6 + 6·(√2)) cm

The perimeter of the of the shaded region = (3·π + 6 + 6·(√2)) cm

Part B

Given that the figure is made up of portion of a square and a semicircle, we have;

BC ≅ AB = 6 cm

The area of semicircle BC with radius BC/2 = 3 is 1/2×π×r² = 1/2×π×3² = 4.5·π cm²

Triangle ABC = 1/2 × Area of square  from which ABC is cut

The area of triangle ABC = 1/2×Base ×Height = 1/2×AB×BC = 1/2×6×6 = 18 cm²

The area of the figure = The area of semicircle BC + The area of triangle ABC

The area of the figure = 4.5·π cm² + 18 cm² =

sss (side, side, side)

sas (side, angle, side)

asa (angle, side, angle)

aas (angle, angle, side)

hl (hypotenuse, leg)