The longer leg of a right triangle is 3 cm longer than the shorter leg.?

Let x be the length of the shorter leg
then hypotenuse = x+6
longer leg = x+3
and... the pythagoras theorem
x^2 + (x+3)^2 = (x+6)^2
==> x^2 + x^2 + 6x + 9 - x^2 -12x -36 = 0
==> x^2 -6x -27 = 0
==> x = (6 +/- √(36 + 108))/2 = (6+/-√144)/2 = 18/2 = 9
so.. 9, 12, 15

The longer leg of a right triangle is 3 cm longer than the shorter leg:

Let x = length of shorter leg

Therefore, (x + 3) is the length of the longer leg.

The hypotenuse is 6 cm longer than the shorter leg, so ..

(x + 6) is the hypotenuse.

Now that we have an expression for all three sides, we can put this through Pythagorean Theorem and solve for x:

a² + b² = c²
x² + (x + 3)² = (x + 6)²

simplify each side and put resulting quadratic into standard form:

x² + x² + 6x + 9 = x² + 12x + 36
2x² + 6x + 9 = x² + 12x + 36
x² - 6x - 27 = 0

This factors ...

(x - 9)(x + 3) = 0
x = -3 and 9

We'll throw out the negative number, and we have:

x = 9

So 9 cm is the shortest leg, followed by 12cm, then 15cm for the hypotenuse.

Pythagoras theorem ---> 9 , 12 , 15cm