The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the other two sides of the triangle is 5 cm. Find the lengths of these sides.
Let the length of one side of right triangle be $=x \mathrm{~cm}$ then other side be $=(x+5) \mathrm{cm}$
And given that hypotenuse $=25 \mathrm{~cm}$
As we know that by Pythagoras theorem,
$x^{2}+(x+5)^{2}=(25)^{2}$
$x^{2}+x^{2}+10 x+25=625$
$2 x^{2}+10 x+25-625=0$
$2 x^{2}+10 x-600=0$
$x^{2}+5 x-300=0$
$x^{2}-15 x+20 x-300=0$
$x(x-15)+20(x-15)=0$
$(x-15)(x+20)=0$
So, either
$(x-15)=0$
$x=15$
Or
$(x+20)=0$
$x=-20$
But the side of right triangle can never be negative
Therefore, when $x=15$ then
$x+5=15+5$
$=20$
Hence, length of one side of right triangle be $=15 \mathrm{~cm}$ then other side be $=20 \mathrm{~cm}$
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