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ICSE Class 9 Answered

Perpendiculars are drawn from a point within an equilateral triangle to the three sides. Prove that sum of three perpendiculars is equal to the altitude of the triangle.
Asked by nehahemaraj14 | 06 Jan, 2019, 08:40: PM
answered-by-expert Expert Answer

As given in the question, perpendiculars OD, OE and OF are drawn from O to sides AB, BC and AC of the equlateral triangle ABC
 
Area of ΔOBC = (1/2)×BC×OE ...................(1)
Area of ΔOAC = (1/2)×AC×OF ...................(2)
Area of ΔOAB = (1/2)×AB×OD....................(3)
 
Let sides of ΔABC be a
 
using (1), (2) and (3), Area of ΔABC = Area of ΔOBC +Area of ΔOBC +Area of ΔOBC = (1/2)×a×( OD+OE+OF) ...................(4)
 
if h is the altitude of ΔABC, then area of ΔABC = (1/2)×a×h  ............................(5)
 
hence from (4) and (5), we can say sum of perpendiculars ( OD+OE+OF) is equal to altitude h
Answered by Thiyagarajan K | 06 Jan, 2019, 09:06: PM

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