Find the value of x in the right triangle if the sides given are x, x+1 and 7.
assuming 7 is the hypotenuse, we have
x^2 + (x+1)^2 = 7^2
Otherwise, we must have
x^2 + 7^2 = (x+1)^2
so, pick your poison and find x.
To find the value of x in the right triangle, we can use the Pythagorean theorem.
The Pythagorean theorem states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the sides given are x, x+1, and 7, with x being the length of one of the legs, x+1 being the length of the other leg, and 7 being the length of the hypotenuse.
So, we can set up the equation based on the Pythagorean theorem:
x^2 + (x+1)^2 = 7^2
Now, we can expand and simplify the equation:
x^2 + (x^2 + 2x + 1) = 49
Combine like terms:
2x^2 + 2x + 1 = 49
Rearrange the equation to get all the terms on one side:
2x^2 + 2x - 48 = 0
Now we can solve this quadratic equation for x. We can either factor it or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 2, and c = -48. Plugging these values into the quadratic formula:
x = (-2 ± √(2^2 - 4(2)(-48))) / (2(2))
Simplifying:
x = (-2 ± √(4 + 384)) / 4
x = (-2 ± √388) / 4
Now, we can simplify further by finding the square root of 388:
x = (-2 ± √(4 * 97)) / 4
x = (-2 ± 2√97) / 4
Finally, we can simplify further by factoring out a 2:
x = 2(-1 ± √97) / 4
Now, we can divide both the numerator and denominator by 2:
x = (-1 ± √97) / 2
So, the two possible values for x in the right triangle are (-1 + √97) / 2 or (-1 - √97) / 2.