Opposite & Adjacent Sides in a Right Angled Triangle
In a triangle...
Question
In a triangle PQR, let ∠PQR=30∘ and the sides PQ and QR have lengths 10√3 and 10, respectively. Then, which of the following statement(s) is (are) TRUE?
A
∠QPR=45∘
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B
The area of the triangle PQR is 25√3 and ∠QRP=120∘
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C
The radius of the incircle of the triangle PQR is 10√3−15
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D
The area of the circumcircle of the triangle PQR is 100π
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Solution
The correct option is D The area of the circumcircle of the triangle PQR is 100π
By cosine law, cos30∘=100+300−(PR)22×10×10√3 ⇒PR=10 ⇒PR=QR,then∠QPR=∠PQR ∴∠QPR=30∘and∠QRP=120∘ area(PQR),△=12×10√3×10sin30∘=25√3
Radius of incircle is given by r=△S=25√312(10+10+10√3)=10√3−15
Radius of the circumcircle is R=PR2sin30∘=10
Therefore, area of the circumcircle =π(10)2=100π