Answer:
Side of equilateral Triangle =12 inch
Let r be the radius of Smallest circle which fits inside the triangle.
Theorem Which will be Used
→The Point of intersection of end point of radius and tangent through that end point makes an angle of 90°.
→Angle Subtended by an arc at the center is twice the angle subtended at any point on the circle.
→Perpendicular from the center of the circle to the chord bisects the chord.
→ Area of Equilateral Triangle
→Area of Triangle
⇒ Radius of the smallest circle
→ Area of Equilateral Triangle having side 12 inches
------------------------------(1)
→Area of 3 Triangle having base =12 inches and height =r inches
-----------------------------(2)
Equating (1) and (2), that is both the Areas are equal.
⇒Radius of the largest circle
Each Interior angle of equilateral Triangle=60°
∠AOC=2×60°=120°-----Angle Subtended by an arc at the center is twice the angle subtended at any point on the circle.
r=Radius of circle
→∠AOC+∠CAO+∠ACO=180°---Angle sum property of triangle.
→ 120°+2∠CAO=180°------[OA=OC, →∠CAO=∠OCA------Angle Opposite to equal side of a triangle are equal.]
→2∠CAO=180°-120°
→2∠CAO=60°
→∠CAO=30°-------Dividing both sides by, 2
→∠MAO+∠AMO+∠AOM=180°---Angle sum property of triangle.
→∠MAO+90°+60°=180°
→∠MAO=180°-150°
→∠MAO=30°
In ΔOMA
AC=12 inches
AM=MC=6 inches--Perpendicular from the center of the circle to the chord bisects the chord.
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