Answer:
Exact area = 48√3 - 16π square units
Rounded area = 32.87 square units (nearest hundredth)
Step-by-step explanation:
In a circle, if two tangent segments are drawn from an exterior point to the circle, the lengths of these tangent segments will be congruent. Therefore, assuming that points A and B on the circumference of the circle are points of tangency, both tangent segments measure 12 units.
To find the area of the shaded region, we need to subtract the area of the sector of the circle subtended by points A and B from the area of the quadrilateral (kite) formed by the center of the circle, the points of tangency (A and B), and the exterior point of intersection of the tangent segments.
Label the exterior point of intersection of the two tangent segments as point C. Add the center of the circle and label it point D. Draw two line segments from points A and B to the center D. These are the radii of the circle. The two radii and the two tangent segments form kite ABCD.
If two tangent segments are drawn to the circle from one exterior point, the measure of the angle formed by the two lines is half of the (positive) difference of the measures of the intercepted arcs. As minor arc AB measures 120°, then major arc AB must measure 240°. Therefore, angle ACB between the two tangent segments is:
[tex]m\angle ACB = \dfrac{240^{\circ}-120^{\circ}}{2}=60^{\circ}[/tex]
The degree measure of an arc is equal to the measure of the central angle subtended by the arc. Therefore, as arc AB measures 120°, the central angle of the sector of the circle ABD also measures 120°.
If we connect points C and D with a line segment, we create two congruent right 30-60-90 triangles, with the longest leg measuring 12 units and the shortest leg representing the radius of the circle.
In a 30-60-90 triangle, the shortest leg is 1/√3 times the length of the longest leg. Therefore, the radius of the circle is:
[tex]r = \dfrac{1}{\sqrt{3}} \cdot 12\\\\\\r = \dfrac{12}{\sqrt{3}}\\\\\\r = \dfrac{12\cdot \sqrt{3}}{\sqrt{3}\cdot \sqrt{3}}\\\\\\r= \dfrac{12\sqrt{3}}{3}\\\\\\r=4\sqrt{3}[/tex]
Now we have the radius r = 4√3 and the central angle 120°, we can find the area of sector ABD using the area of a sector formula:
[tex]A=\left(\dfrac{\theta}{360^{\circ}}\right) \pi r^2\\\\\\A=\left(\dfrac{120^{\circ}}{360^{\circ}}\right) \pi \cdot \left(4\sqrt{3}\right)^2\\\\\\A=\dfrac{1}{3} \pi \cdot 48\\\\\\A=16\pi\\\\\\[/tex]
Therefore, the exact area of sector ABD is 16π square units.
To find the area of a kite, multiply the side lengths of the two non-congruent sides by the sine of the angle between those sides. As tangent lines to a circle are perpendicular to the radius drawn to the point of tangency, the angle between the non-congruent sides is 90°. Therefore:
[tex]\textsf{Area of kite $ABCD$}=AC \cdot AD \cdot \sin CAD\\\\\textsf{Area of kite $ABCD$}=12 \cdot 4\sqrt{3} \cdot \sin 90^{\circ}\\\\\textsf{Area of kite $ABCD$}=12 \cdot 4\sqrt{3} \cdot 1\\\\\textsf{Area of kite $ABCD$}=48\sqrt{3}[/tex]
Finally, to determine the area of the shaded region, we subtract the area of sector ABD from the area of kite ABCD:
[tex]\textsf{Area of shaded region}=\textsf{Kite $ABCD$} - \textsf{Area of sector $ABD$}\\\\\textsf{Area of shaded region}=48\sqrt{3}-16\pi\\\\\textsf{Area of shaded region}=32.87295630...\\\\\textsf{Area of shaded region}=32.87\; \sf square\;units\;(nearest\;hundredth)[/tex]
Therefore, the area of the shaded region is exactly 48√3 - 16π square units, which is approximately 32.87 square units rounded to the nearest hundredth.
