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Question

From a point p, inside of an equilateral triangle ABC, the perpendicular distances of the three sides are 63 cm,93 cm and 122 cm, respectively. Find the semi perimeter of the triangle.
1227459_0a7b555aecc9453fb77d0d3d836f4dd1.png

A
81 cm
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B
54 cm
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C
108 cm
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D
162 cm
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E
none of these
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Solution

The correct option is A 81 cm
Given 63cm,93cm,123cm are the three perpendiculars from the point O which lies inside the equilateral triangle.

Let P1=63cm,P2=93cm; and P3=123cm.

If we consider the triangles APB,APC,BPC, we can see that the 3 triangles will be equal to total area of triangle ABC
Area (ABC)=Area(APB)+Area(APC)+Area(BPC)
34a2=12(93+123+63)×a

32.a=(6+9+12)3cm

3a=273×2

a=54cm.

Now perimeter of equilateral triangle.
=3a

=3×54

=162cm

In this case

Semi-perimeter of equilateral triangle.
=12×162
=81cm


1511900_1227459_ans_c6e39e6283924883860e1f9947c57080.png

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