a. the area of the shaded region is 8.1879 cm²
b. the height of the triangle is 10.0708 cm
Since the shaded region is a segment, we need to find the area of the shaded region.
How to find the area of the shaded region?
Area of the shaded region A = area of sector, A' - area of triangle, A"
The area of sector
Area of sector, A' = Ф/360 × πr² where
- Ф = angle of sector = 46° and
- r = radius of circle = 14 cm
So, substituting the values of the variables into the equation, we have
A' = Ф/360 × πr²
= 46/360 × π(14 cm)²
= 46/360 × 196πcm²
= 9016π/360 cm²
= 25.0444π cm²
A' = 78.6793 cm²
So, the area of the sector is 78.6793 cm²
The Area of triangle
Area of triangle, A" = 1/2r²sinФ where
- r = radius of circle = 14 cm and
- Ф = angle of sector = 46°
So, substituting the values of the variables into the equation, we have
A" = 1/2r²sinФ
= 1/2 × (14 cm)²× sin46°
= 1/2 × 196 cm² × 0.7193
= 98 cm² × 0.7193
A" = 70.4914 cm²
a. The Area of the shaded region
The area of the shaded region A = area of sector A' - area of triangle, A"
= 78.6793 cm² - 70.4914 cm²
= 8.1879 cm²
So, the area of the shaded region is 8.1879 cm²
b. What is the height of the triangle?
Using SOHCAHTOA in the triangle,
sin46° = h/14 cm
So, h = 14 cm × sin46°
= 14 cm × 0.7193
= 10.0708 cm
So, the height of the triangle is 10.0708 cm
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