Triangle pqr is such that pq=6cm <rpq=50degrees <pqr=60 degrees calculate the area of a circle passing through vertices p'q'rlte the area
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Answer
The area of the circle passing through the vertices P, Q, and R is approximately 32.0 cm².
Step-by-step solution
4 stepsHere's how to calculate it:
Find the third angle of the triangle (∠R)
The sum of angles in a triangle is 180°.
∠R = 180° - (∠P + ∠Q)
∠R = 180° - (50° + 60°)
∠R = 180° - 110°
∠R = 70°Use the Law of Sines to find the circumradius (R)
The Law of Sines states that for a triangle with sides a, b, c and opposite angles A, B, C, the ratio a/sin A = b/sin B = c/sin C is equal to 2R, where R is the circumradius.
We have side PQ (let's call it 'r') = 6 cm, and its opposite angle is ∠R = 70°.
r / sin(R) = 2R
6 / sin(70°) = 2R
R = 6 / (2 * sin(70°))
R = 3 / sin(70°)Calculate the value of R
sin(70°) ≈ 0.9396926
R ≈ 3 / 0.9396926
R ≈ 3.192537 cmCalculate the area of the circle
The area of a circle is given by the formula A = πR².
A = π * (3.192537)²
A = π * 10.192297
A ≈ 3.14159265 * 10.192297
A ≈ 31.9599 cm²Rounding to three significant figures, the area is 32.0 cm².
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