Answer:
Exact area = (18√3 - 6π) cm²
Rounded area = 12.33 cm² (2 d.p.)
Step-by-step explanation:
The tangent of a circle is always perpendicular to the radius. Therefore, triangle ATO is a right triangle, where ∠ATO = 90°.
Given:
- Radius = OT = 6 cm
- OA = 12 cm
Considering that triangle ATO is a right triangle where its hypotenuse (OA) is twice the length of its shortest leg (OT), it is a special 30-60-90 right triangle. In this type of triangle, the legs are in the ratio 1 : √3 : 2, so AT = 6√3 cm, and the angle opposite the longest leg is 60°. Therefore, the measure of the central angle TOA is 60°.
To find the area of the shaded region, we can subtract the area of the sector of the circle from the area of right triangle ATO:
[tex]\text{Area of shaded region}=\text{Area of $\triangle ATO$}-\text{Area of sector}[/tex]
The area of right triangle ATO can be found by halving the product of its two legs, OT and AT:
[tex]\text{Area of $\triangle ATO$}=\dfrac{1}{2}\cdot OT \cdot AT\\\\\\\text{Area of $\triangle ATO$}=\dfrac{1}{2}\cdot 6 \cdot 6\sqrt 3\\\\\\\text{Area of $\triangle ATO$}=18\sqrt 3\; \sf cm^2\\\\\\[/tex]
The area of the sector can be found by using the area of a sector formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Area of a sector}}\\\\A=\left(\dfrac{\theta}{360^{\circ}}\right) \pi r^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the radius.}\\\phantom{ww}\bullet\;\;\textsf{$\theta$ is the angle measured in degrees.}\end{array}}[/tex]
In this case:
- r = OT = 6 cm
- θ = ∠TOA = 60°
Substitute the values into the formula and solve for area:
[tex]\text{Area of sector}=\left(\dfrac{60^{\circ}}{360^{\circ}}\right) \pi \cdot 6^2\\\\\\\text{Area of sector}=\dfrac{1}{6} \pi \cdot 36\\\\\\\text{Area of sector}=6 \pi \;\sf cm^2\\\\\\[/tex]
Finally subtract the area of the sector from the area of the triangle to find the area of the shaded region:
[tex]\text{Area of shaded region}=\text{Area of $\triangle ATO$}-\text{Area of sector}\\\\\text{Area of shaded region}=18\sqrt{3}-6\pi\\\\\text{Area of shaded region}=12.33\; \sf cm^2\;(2\;d.p.)[/tex]
Therefore, the exact area of the shaded region is (18√3 - 6π) cm², which is approximately 12.33 cm² (rounded to two decimal places).
