Mathematics - Properties of Triangle Question with Solution | TestHub

MathematicsProperties of TriangleLength of angle bisector and median / Dist. between sp. PointsMedium2 minQB

MathematicsMediumsingle choice

In a scalene triangle ABC, D is on AB with $C D^{2}$ =AD·DB. If sin A·sin B= $sin^{2} (C /2)$ , what is CD?

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Answer:

Solution:

Put $α$ = $∠ A C D$ , $β$ = $∠ D CB$ , so $α$ + $β$ =C. By sine rule in ACD and BCD, AD/CD=sin $α$ /sin A and DB/CD=sin $β$ /sin B. Since AD·DB= $C D^{2}$ , sin $α$ sin $β$ =sin A sin B= $sin^{2} (C /2)$ . But sin $α$ sin $β$ =[cos( $α$ - $β$ )-cos C]/2≤[1-cos C]/2= $sin^{2} (C /2)$ , with equality only $α$ = $β$ . Therefore CD bisects angle C.

Stream:JEESubject:MathematicsTopic:Properties of TriangleSubtopic:Length of angle bisector and median / Dist. between sp. Points

⏱ 2mℹ️ Source: QB

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