Two circles with equal radii, r = 9 ext cm. Centers A and B. Angle CAD = 80^. i) Calculate the area of Sector…

Question

Two circles with equal radii, r = 9 ext cm. Centers A and B. Angle CAD = 80^. i) Calculate the area of Sector CAD. ii) Calculate the area of Triangle CAD.

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1) Geometry Problem

Given:

Two circles with equal radii, $r = 9 \text{ cm}$ .
Centers A and B.
Angle $\angle CAD = 80^\circ$ .
i) Calculate the area of Sector CAD.

Identify the formula for the area of a sector.
The area of a sector is given by the formula:
$A_{\text{sector}} = \frac{\theta}{360^\circ} \pi r^2$
where $\theta$ is the central angle in degrees and $r$ is the radius.
Substitute the given values into the formula.
Given $\theta = 80^\circ$ and $r = 9 \text{ cm}$ .
$A_{\text{sector}} = \frac{80^\circ}{360^\circ} \times \pi \times (9 \text{ cm})^2$
Simplify the expression.
$A_{\text{sector}} = \frac{2}{9} \times \pi \times 81 \text{ cm}^2$ $A_{\text{sector}} = 2 \times 9 \times \pi \text{ cm}^2$ $A_{\text{sector}} = 18\pi \text{ cm}^2$
Using $\pi \approx 3.14159$ :
$A_{\text{sector}} \approx 18 \times 3.14159 \text{ cm}^2$ $A_{\text{sector}} \approx 56.55 \text{ cm}^2$
The area of Sector CAD is $\boxed{18\pi \text{ cm}^2 \approx 56.55 \text{ cm}^2}$ .
- ii) Calculate the area of Triangle CAD.
Identify the formula for the area of a triangle given two sides and the included angle.
Triangle CAD is an isosceles triangle with sides $AC = AD = r = 9 \text{ cm}$ and the included angle $\angle CAD = 80^\circ$ .
The area of a triangle is given by the formula:
$A_{\text{triangle}} = \frac{1}{2}ab\sin C$
where $a$ and $b$ are the lengths of two sides and $C$ is the included angle.
Substitute the given values into the formula.
Given $a = 9 \text{ cm}$ , $b = 9 \text{ cm}$ , and $C = 80^\circ$ .
$A_{\text{triangle}} = \frac{1}{2} \times 9 \text{ cm} \times 9 \text{ cm} \times \sin(80^\circ)$
Simplify the expression.
$A_{\text{triangle}} = \frac{81}{2} \sin(80^\circ) \text{ cm}^2$ $A_{\text{triangle}} = 40.5 \sin(80^\circ) \text{ cm}^2$
Using $\sin(80^\circ) \approx 0.9848$ :
$A_{\text{triangle}} \approx 40.5 \times 0.9848 \text{ cm}^2$ $A_{\text{triangle}} \approx 39.88 \text{ cm}^2$
The area of Triangle CAD is $\boxed{40.5 \sin(80^\circ) \text{ cm}^2 \approx 39.88 \text{ cm}^2}$ .
- iii) Calculate the area of the shaded region.
Understand the shaded region.
The shaded region is the area of intersection of the two circles (the lens shape). This area is composed of two identical segments, one from each circle. The area of one segment (e.g., from circle A, formed by chord CD) is the area of Sector CAD minus the area of Triangle CAD.
Calculate the area of one segment.
$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}$ $A_{\text{segment}} = 18\pi \text{ cm}^2 - 40.5 \sin(80^\circ) \text{ cm}^2$ $A_{\text{segment}} \approx 56.55 \text{ cm}^2 - 39.88 \text{ cm}^2$ $A_{\text{segment}} \approx 16.67 \text{ cm}^2$
Calculate the total shaded area.
Since the two circles are identical and the intersection is symmetrical, the total shaded area is twice the area of one segment.
$A_{\text{shaded}} = 2 \times A_{\text{segment}}$ $A_{\text{shaded}} = 2 \times (18\pi - 40.5 \sin(80^\circ)) \text{ cm}^2$ $A_{\text{shaded}} = (36\pi - 81 \sin(80^\circ)) \text{ cm}^2$
Using the approximate values:
$A_{\text{shaded}} \approx 2 \times 16.67 \text{ cm}^2$ $A_{\text{shaded}} \approx 33.34 \text{ cm}^2$
The area of the shaded region is $\boxed{(36\pi - 81 \sin(80^\circ)) \text{ cm}^2 \approx 33.34 \text{ cm}^2}$ .

2) Second Problem

The second problem is incomplete in the image provided. Please send the full question and diagram for "2) The figure shows a circle $\angle QOT$ is the diameter and $\angle SRT = 37^\circ$ ".

Two circles with equal radii, r = 9 ext cm. Centers A and B. Angle CAD = 80^. i) Calculate the area of Sector CAD. ii) Calculate the area of Triangle CAD.

Step-by-step solution

1) Geometry Problem

Identify the formula for the area of a sector.

Substitute the given values into the formula.

Simplify the expression.

Identify the formula for the area of a triangle given two sides and the included angle.

Substitute the given values into the formula.

Simplify the expression.

Understand the shaded region.

Calculate the area of one segment.

Calculate the total shaded area.

2) Second Problem

Still stuck on this one?

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