Question

oac is a sector of a circle of radius 8cm, centre O ba and bc are tangents to the circle angle aoc = 120 degrees calculate the area of the shaded region give your answer to 3 significant figures

Answer 1

Answer:

The area of the shaded region is $49.5 cm^{2}$ to 3. significant figures

Step-by-step explanation:

Step one: find the two diagonals of the kite.

The Horizontal diagonal can be obtained using the cosine rule:

$AC^{2}=OA^{2}-OC^{2}- 2\times (OA)\times (OC)cos (\theta)$

$AC^{2}=8^{2}+8^{2}- 2\times 8 \times 8 \times cos (120)= 212\\AC =\sqrt{212}=14.56cm$

The vertical diagonal of the kite can be obtained by Pythagoras' Theorem:

Please note the law in circle geometry which states that a radius and a tangent always meet at right angles.

This implies that triangle OBC is a right-angled triangle, with angle OCB being 90 degrees, and COB being 60 degrees. This is because the diagonal divides the 120-degree angle into half.

$cos 60 = \frac{8}{OB}\\OB=\frac{8}{cos 60}= 16cm$

Step two: Use the dimensions of the two diagonals of the Kite to find the area:

The area of a Kite is obtained using this formula:

$Area = \frac{pq}{2}$, where p and q are the two diagonals.

hence,

$Area = \frac{14.56 \times 16}{2}= 116.48cm^{2}$

Step three: Calculate the area of the sector of the circle.

Area of the sector is obtained using this formula

$Area = \frac{\theta}{360}\times \pi\times r^{2}\\Area = \frac{120}{360} \times \pi \times 8^{2}=67.03cm^{2}$

Step Four: Subtract the area of the sector from the area of the kite:

Area of the shaded region will be $116.48cm^{2} -67.03cm^{2} = 49.45cm^{2}$

This will be$49.5 cm^{2}$ to 3. significant figures

Answer 2

Answer:

43.8 cm²

Step-by-step explanation:

1. Split the kite in two and find the length of OB

When split the kite in two, you will find that the 120° becomes 60°, as they are divided into two.

Then, use the sine rule we can find out the length of OB. It goes like this:

$\frac{OB}{sin90} = \frac{8}{sin30}\\ \\{OB} = \frac{8sin90}{sin30}\\\\ {OB} = 16cm$

2. Calculate the area of the kite

Since we have split the kite into two triangles and we know two side of the triangles, then we can calculate the area of one triangle using the formula of $area of triangle=\frac{1}{2}absinc$.

$\frac{1}{2}$×8×16×sin60 = $32\sqrt{3}$

But as there are two triangles, we duplicate the answer.

2×$32\sqrt{2}$ = $64\sqrt{3}$

2. Calculate the area of the sector

area of sector =$\frac{angle of sector}{360}$×πr²

$\frac{120}{360}$×8²π = $\frac{64}{3}$π

3. Subtract the area of the sector from the area of the kite

$64\sqrt{3}$ - $\frac{64}{3}$π = 43.83060841 ≈ 43.8 cm² (to 3 s.f.)

HOPE THIS WILL HELP YOU :)