Question 1207560
<pre>
The sides of a right angled triangle are such that the sum of the length of the
longest and that of the shortest side is twice the length of remaining side, <font color="red"><b>what
is the length of</font></b> the longest side of the triangle if the longer of the sides
containing the right angle is 9 CM more than half the hypotenuse is??

Let the length of the hypotenuse (the longest side) be x.
Let the length of the shortest side be y.
Then the length of the remaining side is {{{sqrt(x^2-y^2)}}}

{{{drawing(200,160,-.5,4.5,-.8,3.2,
locate(1.8,0,sqrt(x^2-y^2)),locate(4.1,1.5,y),locate(1.8,1.9,x),


triangle(0,0,4,0,4,3)  )}}}



{{{x+y}}}{{{""=""}}}{{{2*sqrt(x^2-y^2)}}}

{{{x+y}}}{{{""=""}}}{{{2*sqrt((x-y)(x+y))}}}

{{{x+y}}}{{{""=""}}}{{{2*sqrt(x-y)*sqrt(x+y))}}}

{{{(x+y)/sqrt(x+y)}}}{{{""=""}}}{{{2*sqrt(x-y)}}} 

{{{sqrt(x+y)}}}{{{""=""}}}{{{sqrt(4)*sqrt(x-y)}}} 

{{{sqrt(x+y)}}}{{{""=""}}}{{{sqrt(4(x-y))}}}

What's under the radicals must be equal

{{{x+y}}}{{{""=""}}}{{{4(x-y)}}}

{{{x+y}}}{{{""=""}}}{{{4x-4y}}}

{{{5y}}}{{{""=""}}}{{{3x}}}

So {{{y}}}{{{""=""}}}{{{expr(3/5)x}}}

Now the sides of the right triangle are:

The length of the hypotenuse (the longest side) is x.
The length of the shortest side is {{{expr(3/5)x}}}
Then the length of the remaining side is 
{{{sqrt(x^2-y^2)}}}{{{""=""}}}{{{sqrt(x^2-(expr(3/5)x)^2)}}}{{{""=""}}}{{{sqrt(x^2-expr(9/25)x^2)}}}{{{""=""}}}{{{sqrt(expr(16/25)x^2)}}}{{{""=""}}}{{{expr(4/5)x}}}

{{{drawing(200,160,-.5,4.5,-.8,3.2,
locate(1.8,0,expr(4/5)*x),locate(4.1,1.5,expr(3/5)*x),locate(1.8,1.9,x),


triangle(0,0,4,0,4,3)  )}}}


Since the longer of the sides containing the right
angle is 9 CM more than half the hypotenuse,

{{{expr(4/5)x}}}{{{""=""}}}{{{expr(1/2)x+9}}}
{{{8x}}}{{{""=""}}}{{{5x+90}}}
{{{3x}}}{{{""=""}}}{{{90}}}
{{{x}}}{{{""=""}}}{{{30}}}

So the hypotenuse is 30 cm in length.

{{{drawing(200,160,-.5,4.5,-.8,3.2,
locate(1.8,0,24),locate(4.1,1.5,18),locate(1.8,1.9,30),


triangle(0,0,4,0,4,3)  )}}}

Edwin</pre>