Answer :
Answer:
182 cm²
Step-by-step explanation:
Given that points R and T lie on the circumference of circle O, line segment RT forms a chord of circle O.
Given that point M is the midpoint of RT, it follows that RM = MT.
Since RT = 12 cm, then RM = MT = 6 cm.
As point M is the midpoint of RT, then line segment OM is perpendicular to chord RT. Therefore, triangle OMR is a right triangle, where:
- RM = 6
- m∠ROM = 52°
To determine the area of circle O, we first need to find its radius (r).
Since the radius (r) is the hypotenuse (OR) of right triangle OMR, and we have the measure of angle ROM and the side opposite this angle, we can use the sine trigonometric ratio to determine the length of the hypotenuse (radius OR):
[tex]\sin \theta=\dfrac{\sf opposite\;side}{\sf hypotenuse} \\\\\\ \sin ROM=\dfrac{RM}{OR} \\\\\\ \sin 52^{\circ}=\dfrac{6}{r} \\\\\\ r=\dfrac{6}{\sin 52^{\circ}}[/tex]
Therefore, the radius (r) of circle O is:
[tex]r=\dfrac{6}{\sin 52^{\circ}}[/tex]
To find the area of the circle, substitute the radius into the area of a circle formula, A = πr²:
[tex]A=\pi \cdot \left(\dfrac{6}{\sin 52^{\circ}}\right)^2 \\\\\\ A=\pi \cdot 57.9746602866...\\\\\\A=182.13276685104...\\\\\\A=182\; \sf cm^2\;(3\;s.f.)[/tex]
Therefore, the area of the circle correct to 3 significant figures is:
[tex]\LARGE\boxed{\boxed{182\; \sf cm^2}}[/tex]
