A right triangle have sides a,b and hypotenuse h. A square inscribed in triangle having sides on a, b and touches h. Find length of side of square in terms of a ,b.

• area of triangle=ab/2
  ab/2=area of 2 smaller triangle + area of square
  let side of square is x
  then
  (a-x)*x/2+(b-x)*x/2+x^2=ab/2
  after solve this we get x=ab/(a+b)
  so side of square is x= ab/(a+b)
• 10 years agoHelpfull: Yes(63) No(0)
• In a triangle, if square is inscribed such that it touches the hypotenuse will make 2 smaller triangles and the square itself. (here in this problem)
  the area of the bigger triangle is 1/2*a*b (1) ( a is base and b is height)
  area of square is x^2 ( let x be the side of square)
  so the area of 1st smaller triangle is = 1/2 *x * (b-x) ( this smaller triangle will have base as the side of the square and height is total height of the bigger triangle minus the side of the square)
  the area of the 2nd smaller triangle is = 1/2 *(a-x)*x ( this smaller triangle will have height equal to the side of the square and base equal to the total base (which is a of bigger triangle) minus side of the square..
  now add... the area of two smaller triangles and the square area which will be equal to the total area of the bigger square given by eqn (1)..
  that is .. 1/2 *(a-x)* x + 1/2 *x*(b-x)+x^2=1/2*a*b.
  this will give the square side x=ab/a+b.
• 10 years agoHelpfull: Yes(6) No(0)
• ans is ab/a+b
• 10 years agoHelpfull: Yes(5) No(2)
• h^2=a^2+b^2
  area of triangle = (1/2)*a*b = (1/2)*h*(√2 *x)
  where side of square=x & diagonal= √2*x
  => a*b=√2*h*x
  => x=a*b/√2*h
  => x=ab/sqrt[2(a^2+b^2)]
  
• 10 years agoHelpfull: Yes(4) No(24)
• a/2 , b/2
  all the triangles form inside are congruent.
• 10 years agoHelpfull: Yes(1) No(1)
• side of square is ab/2(a2+b2)
  
• 10 years agoHelpfull: Yes(1) No(2)
• Let side of sq be x. Now this sq lies with rt triangle so it vl divide the hyp into two halves.by pythagpreus theprem we get x=a+b/2
• 10 years agoHelpfull: Yes(1) No(3)
• For getting square inside a should be equal to b.
  Hence a2+b2/ab satisfies all values of integers.
• 10 years agoHelpfull: Yes(1) No(0)
• @aditya how u solvd?
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• answer with similar triangle property and area method come different.
• 10 years agoHelpfull: Yes(0) No(0)