the length of the hypotenuse of a right angled triangle is one unit more than twice the length of shortest side and the other side is one unit less than twice the length of the shortest side. find the lengths of other two sides.
If x is the sort side, then
x^2 + (2x-1)^2 = (2x+1)^2
To find the lengths of the other two sides of the right-angled triangle, let's assume that the shortest side is represented by variable "x".
According to the given information, the length of the hypotenuse is one unit more than twice the length of the shortest side. So, the length of the hypotenuse can be expressed as 2x + 1.
Similarly, the length of the other side (adjacent to the right angle) is one unit less than twice the length of the shortest side. Therefore, the length of this side can be written as 2x - 1.
We have now expressed the lengths of the other two sides in terms of "x". By substituting these expressions into Pythagoras' theorem, we can solve for "x" and find the lengths of the other two sides.
According to Pythagoras' theorem:
(hypotenuse)^2 = (side 1)^2 + (side 2)^2
Substituting the values we obtained earlier, we get:
(2x + 1)^2 = x^2 + (2x - 1)^2
Expanding and simplifying the equation:
4x^2 + 4x + 1 = x^2 + 4x^2 - 4x + 1
Combine like terms:
4x^2 + 4x + 1 = 5x^2 - 4x + 1
Rearrange the equation:
0 = 5x^2 - 4x - 4x^2
Combine like terms:
0 = x^2 - 8x
Factor out "x":
0 = x(x - 8)
From this equation, we have two possible solutions:
1) x = 0
2) x - 8 = 0, which gives x = 8
Since the length of a side cannot be zero in a triangle, we discard the first solution, x = 0.
Therefore, the length of the shortest side is x = 8.
Now, substituting this value of x into the expressions we obtained earlier, we can find the lengths of the other two sides:
Length of hypotenuse = 2x + 1 = 2(8) + 1 = 17 units
Length of other side = 2x - 1 = 2(8) - 1 = 15 units
Hence, the lengths of the other two sides are 17 units and 15 units, respectively.