suppose that the length of one leg of a right triangle is 2 inches less than the length of the other leg.the length of hypotenuse is 10 inches.find the length of each leg.
Just good ol' Pythagoras here ...
x^2 + (x-2)^2 = 10^2
x^2 + x^2 - 4x + 4 - 100 = 0
2x^2 - 4x - 96 = 0
x^2 - 2x - 48 = 0
(x - 8)(x + 6) = 0
x = 8 or x = -6, but we can't have a negative side
so the triangle is 8 by 6 by 10
(which you might recognize as double the well-known 3-4-5 right angled triangle)
To find the length of each leg of the right triangle, we can use the Pythagorean theorem.
The Pythagorean theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's call the length of one leg "x" and the length of the other leg "x + 2".
According to the problem, the length of the hypotenuse is 10 inches.
Now, we can construct the equation based on the Pythagorean theorem:
x^2 + (x + 2)^2 = 10^2
Expanding the equation:
x^2 + x^2 + 4x + 4 = 100
Combining like terms:
2x^2 + 4x - 96 = 0
Dividing the equation by 2:
x^2 + 2x - 48 = 0
Now, we can solve this quadratic equation to find the value of x.
Using factoring or the quadratic formula, we find that x = 6 or x = -8.
Since we are dealing with lengths, x cannot be negative, so we discard x = -8.
Therefore, the length of one leg is x = 6 inches.
The length of the other leg is x + 2 = 6 + 2 = 8 inches.
So, the lengths of the legs of the right triangle are 6 inches and 8 inches.