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CBSE Class 10 Answered

  1. If  A be the area of a right triangle and x is one of the sides containing right angle .Prove that the length  of altitude on the hypotenuse is 2Ax / √(x^2 +4A^2).
Asked by sucharitasahoo1 | 24 Nov, 2017, 06:04: PM
answered-by-expert Expert Answer
begin mathsize 16px style Base space of space the space right space angled space triangle space is space apostrophe straight b apostrophe space units. space
Area space of space the space right space angled space triangle space is space " straight A apostrophe space sq space units. space
straight A space equals space 1 half space cross times space straight b space cross times space straight h space
rightwards double arrow space straight h space equals space fraction numerator 2 straight A space over denominator straight b end fraction
so comma another space side space of space the space right space angled space triangle space containing space the space right space angle space equals fraction numerator 2 straight A space over denominator straight b end fraction
Hypotenuse space of space the space right space angled space triangle space according space to space Pythagoras space theorem colon space space
left parenthesis Hypotenuse right parenthesis squared space equals space left parenthesis straight b right parenthesis squared space plus space open parentheses fraction numerator 2 straight A space over denominator straight b end fraction close parentheses squared space space
rightwards double arrow left parenthesis Hypotenuse right parenthesis squared space equals space left parenthesis straight b right parenthesis squared space plus space open parentheses fraction numerator 4 straight A squared space over denominator straight b squared end fraction close parentheses space space space space
rightwards double arrow space Hypotenuse space equals space square root of left parenthesis straight b right parenthesis squared space plus space open parentheses fraction numerator 4 straight A squared space over denominator straight b squared end fraction close parentheses end root
rightwards double arrow space Hypotenuse space equals space square root of space open parentheses fraction numerator left parenthesis straight b right parenthesis to the power of 4 space plus 4 straight A squared space over denominator straight b squared end fraction close parentheses end root
rightwards double arrow space Hypotenuse space equals space 1 over straight b space square root of left parenthesis straight b right parenthesis to the power of 4 space plus 4 straight A squared space end root
Area space of space the space right space angle space considering space hypotenuse space as space the space base. space space
straight A space equals space 1 half space cross times space Hypotenuse space cross times space altitude space on space hypotenuse space space
rightwards double arrow space 2 straight A space equals space space space 1 over straight b space square root of left parenthesis straight b right parenthesis to the power of 4 space plus 4 straight A squared space end root cross times space altitude space on space hypotenuse space
rightwards double arrow space 2 Ab space equals space square root of left parenthesis straight b right parenthesis to the power of 4 space plus 4 straight A squared space end root space cross times space altitude space on space hypotenuse space space
rightwards double arrow space Altitude space on space hypotenuse space equals space fraction numerator 2 Ab over denominator square root of left parenthesis straight b right parenthesis to the power of 4 space plus 4 straight A squared space end root end fraction
Therefore comma space
length space of space the space altitude space on space hypotenuse space of space the space right space angled space triangle space is space fraction numerator 2 Ab over denominator square root of left parenthesis straight b right parenthesis to the power of 4 space plus 4 straight A squared space end root end fraction. end style
Answered by Arun | 24 Nov, 2017, 07:07: PM
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