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In a triangle PQR, let angle PQR=30* and the side PQ and QR have lengths 10root3 and 10 respectively then which of the following ststement(s) is (are) TRUE ? (A) angle QPR=45* (B) the area of the triangle PQR is 25root3 and angle QRP= 120* (C) the radius of the incircle of the triangle PQR is 10root3-15 (D) the area of the circumcircle the triangle PQR is 100 pie
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Asked by ranvirvinay1234 | 08 May, 2020, 19:04: PM
answered-by-expert Expert Answer
Let measure of angle P be x.
Therefore, measure of angle R will be 150-x.
left parenthesis i right parenthesis fraction numerator 10 over denominator sin space x end fraction equals fraction numerator 10 square root of 3 over denominator sin open parentheses 150 minus x close parentheses end fraction rightwards double arrow sin 150 space cos x space minus space cos 150 space sin x equals square root of 3 sin x rightwards double arrow fraction numerator cos x over denominator 2 end fraction plus fraction numerator square root of 3 over denominator 2 end fraction sin x equals square root of 3 sin x rightwards double arrow tan x equals fraction numerator 1 over denominator square root of 3 end fraction rightwards double arrow x equals 30 degree left parenthesis i i right parenthesis A r e a space o f space t r i a n g l e space P Q R space equals space 1 half cross times 10 cross times 10 square root of 3 space sin 30 degree equals 25 square root of 3 angle Q R P equals 150 minus 30 equals 120 degree S i n c e space angle P Q R equals angle Q P R rightwards double arrow P R equals Q R equals 10 left parenthesis i i i right parenthesis I n r a d i u s left parenthesis r right parenthesis equals fraction numerator A r e a space o f space t r i a n g l e over denominator s e m i p e r i m e t e r end fraction equals fraction numerator 25 square root of 3 over denominator 10 plus 5 square root of 3 end fraction equals fraction numerator 25 square root of 3 open parentheses 10 minus 5 square root of 3 close parentheses over denominator 100 minus 75 end fraction equals 10 square root of 3 minus 15 left parenthesis i v right parenthesis A r e a space o f space c i r c u m c i r c l e equals πR squared equals straight pi over 4 open parentheses fraction numerator 10 over denominator 1 divided by 2 end fraction close parentheses squared equals 100 straight pi
Answered by Renu Varma | 11 May, 2020, 00:14: AM

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