Please show solution to this problem.
In triangle A, the longer leg is 6 more than the shorter leg and the hypotenuse is 6 more than the longer leg.
Find the lengths of each side of this right triangle. Show all calculations.
Short leg = _______ Long leg = ______ Hypotenuse = _________
Thank you.
To solve this problem, we can set up equations based on the information given and use algebra to solve for the unknown side lengths. Let's denote the shorter leg as x, the longer leg as x + 6, and the hypotenuse as (x + 6) + 6.
We know that in a right triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the hypotenuse.
Applying this to our triangle, we have:
x^2 + (x + 6)^2 = ((x + 6) + 6)^2
Now let's simplify the equation step by step:
x^2 + (x^2 + 12x + 36) = (x + 12)^2
Expand the squares:
x^2 + x^2 + 12x + 36 = x^2 + 24x + 144
Combine like terms:
2x^2 + 12x + 36 = x^2 + 24x + 144
Move all terms to one side of the equation:
2x^2 + 12x + 36 - x^2 - 24x - 144 = 0
Combine like terms again:
x^2 - 12x - 108 = 0
Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's use factoring.
Factoring the quadratic equation, we have:
(x - 18)(x + 6) = 0
Setting each factor equal to zero, we get:
x - 18 = 0 or x + 6 = 0
Solving these equations, we find:
x = 18 or x = -6
Since we are dealing with the lengths of sides, the solution of x = -6 is not valid, as side lengths cannot be negative.
Therefore, the valid solution is x = 18.
Plugging this value back into our initial equations, we can find the lengths of each side:
Short leg = x = 18 units
Long leg = x + 6 = 18 + 6 = 24 units
Hypotenuse = (x + 6) + 6 = 18 + 6 + 6 = 30 units
So, the lengths of each side of this right triangle are:
Short leg = 18 units
Long leg = 24 units
Hypotenuse = 30 units
x^2 + (x+6)^2 = (x+12)^2
x^2+ x^2+ 12 x + 36 = x^2 +24 x +144
x^2 - 12 x - 108 = 0
108 is 6*18 for example
(x-18)(x+6) = 0
try x = 18
then x+6 = 24
and x+12 = 30
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check
18^2 + 24^2 = 30^2 ????????
324 + 576 = 900 whew !