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Let 'P' be an interior point of Delta ABC. If angle A=45^(@), angle B=60^(@) and angle C=75^(@). If X=area of Delta PBC,Y= area of Delta PAC and Z = area of Delta PAB, then which of the following ratios is/are true ? |
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Answer» If P is the centroid, then X : Y : Z is 1 : 1 : 1  Using properties of median, we have `Delta PBC = Delta PCA = Delta PAB` `therefore Delta PBC : Delta PCA : Delta PAB = 1:1:1` (B)  `Delta PBC : Delta PCA : Delta PAB` `=(1)/(2)ar : (1)/(2)br : (1)/(2)cr` `= a:b:c` `= sin 45^(@): sin 60^(@): sin 75^(@)=2 : sqrt(6):(sqrt(3)+1)` (c )  `Delta PBC : Delta PCA : Delta PAB` `=(1)/(2)a(2R COS B cos C) : (1)/(2)b(2 R cos C cos A) : (1)/(2)c(2R cos A cos B)` = sin A cos B cos C : sin B cos C cos A : sin C cos A cos B `= tan 45^(@) : tan 60^(@) : tan 75^(@)` `= 1: sqrt(3):(2+sqrt(3))` (d)  `Delta PBC : Delta PCA : Delta PAB` `=(1)/(2)R^(2)sin 2A : (1)/(2)R^(2)sin 2B(1)/(2)R^(2)sin 2C` `= sin 2A : sin 2B : sin 2C` `= sin 90^(@) : sin 120^(@) : sin 150^(@)` `= 2 : sqrt(3):1` |
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