Given that:
AB = y - 3
BC = y + 11
AC = 2y
In a right-angled triangle, the hypotenuse (AC) is the longest side.
According to the Pythagorean theorem, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse:
AB^2 + BC^2 = AC^2
Substituting the given values:
(y - 3)^2 + (y + 11)^2 = (2y)^2
Expanding and simplifying:
y^2 - 6y + 9 + y^2 + 22y + 121 = 4y^2
2y^2 + 16y + 130 = 4y^2
-2y^2 + 16y - 130 = 0
Now, we solve this quadratic equation by factoring or using the quadratic formula
Using the quadratic formula:
y = [-b ± sqrt(b^2 - 4ac)] / 2a
a = -2, b = 16, c = -130
y = [-16 ± sqrt(16^2 - 4(-2)(-130))] / 2(-2)
y = [-16 ± sqrt(256 - 1040)] / -4
y = [-16 ± sqrt(-784)] / -4
y = [-16 ± 28i] / -4
y = -4 ± 7i
Therefore, the possible values of y are:
y = -4 + 7i or y = -4 - 7i
Since the length of the sides cannot be imaginary, this means that there is no real solution to the lengths of the sides based on the given information.
In the right angle triangle a b is the height which is y - 3 BC is the length which is y + 11 and ac is the hypotenuse which is 2Y find the length of each side
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