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vladimir1956 [14]
3 years ago
5

The figure above shows a right-angled triangle OAB. AOC is a minor sector enclosed in the triangle. If OA = 7 cm, AB = 6 cm, cal

culate the area and perimeternof the shaded region.​ PLEASE HELP!

Mathematics
1 answer:
dolphi86 [110]3 years ago
7 0

Answer:

Step-by-step explanation:

Given that:

OA = 7 cm, AB = 6 cm. ∠A = 90°, OA = OC = 7 cm

Using Pythagoras theorem: OB² = OA² + AB²

OB² = 6² + 7²=85

OB = √85 = 9.22 cm

to find ∠O, we use sine rule:

\frac{AB}{sin(O)}=\frac{OB}{sin(A)}\\  \\sin(O)=\frac{AB*sin(A)}{OB}=\frac{6*sin(90)}{9.22} =0.65 \\\\O=sin^{-1}0.65=40.6^o

AOC is a minor sector with radius 7 cm and angle 40.6

The Area of the triangle OAB = 1/2 × base × height = 1/2 × OA × AB = 1/2 × 7 × 6 = 21 cm²

Area of sector OAC = \frac{\theta}{360}*\pi r^2=\frac{40.6}{360}*\pi *7^2=17.37 \ cm^2

Area of shaded region = The Area of the triangle OAB - Area of sector OAC = 21 - 17.37 = 3.63 cm²

Perimeter of arc AC = \frac{\theta}{360}*2\pi r=\frac{40.6}{360}*2\pi *7=4.96\ cm

CB = OB - OC = 9.22 - 7 = 2.22

Perimeter of shaded region = AB + CB + arc AC = 6 + 2.22 + 4.96 = 13.18 cm

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