One more triangle question,please.
In this triangle,the short leg is x and the longer leg is 1/2x + 11. If the hypotenuse is 2x + 1, what are the lengths of each side of this right triangle? Please show all calculations.
Short leg = ______ Long leg = ___________ Hypotenuse = ________________
Thank you very much!
This time you try first and show work. Same method.
should look like
11 x^2 - 28 x -480 = 0
(11x+60)(x-8) = 0
To find the lengths of the sides of this right triangle, you need to substitute the given expressions into the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
In this case, the short leg is x, the longer leg is 1/2x + 11, and the hypotenuse is 2x + 1.
So, we can set up the equation:
(x^2) + ((1/2x + 11)^2) = (2x + 1)^2
Simplifying this equation step by step:
(x^2) + ((1/2x + 11)^2) = (2x + 1)^2
(x^2) + ((1/2x + 11)(1/2x + 11)) = (2x + 1)(2x + 1)
Expanding the squares:
(x^2) + (1/4x^2 + 11/2x + 11/2x + 121) = (4x^2 + 2x + 2x + 1)
Combining like terms:
(x^2) + (1/4x^2 + 11x + 11x + 121) = (4x^2 + 4x + 1)
Multiplying all terms by the common denominator (4) to eliminate fractions:
4(x^2) + (x^2 + 44x + 44x + 484) = (4x^2 + 4x + 1)
Distributing:
4x^2 + x^2 + 44x + 44x + 484 = 4x^2 + 4x + 1
Combining like terms:
5x^2 + 88x + 484 = 4x^2 + 4x + 1
Simplifying further:
x^2 + 84x + 483 = 0
Now, we can solve this quadratic equation for x. We can either factor it or use the quadratic formula.
Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 1, b = 84, and c = 483.
Substituting the values into the formula:
x = (-(84) ± √((84)^2 - 4(1)(483))) / (2(1))
Calculating further:
x = (-84 ± √(7056 - 1932)) / 2
x = (-84 ± √5124) / 2
Now you can calculate the value of x and substitute it back into the expressions for the short leg, long leg, and hypotenuse to find their lengths.