If sides of right angled triangle are three consecutive integers.then length of smallest side is
the simplest right-angled triangle has sides
3,4, and 5
If you did not know that,
let the three sides be
x, x+1, and x+2
x^2 + (x+1)^2 = (x+2)^2
x^2 + x^2 + 2x + 1 = x^2 + 4x + 4
x^2 - 2x - 3 = 0
(x-3)(x+1) = 0
x = 3 or x = -1, the last one is silly
x=3
conclusion ....
To determine the length of the smallest side of a right-angled triangle with three consecutive integer sides, we can follow these steps:
Step 1: Understand the problem.
A right-angled triangle has one angle measuring 90 degrees. The sum of the squares of the two shorter sides is equal to the square of the longest side, according to the Pythagorean theorem. We need to find the length of the smallest side.
Step 2: Set up the problem.
Let's assume that the consecutive integers are represented by "x," "x+1," and "x+2" for the smallest, middle, and largest sides, respectively.
Step 3: Apply the Pythagorean theorem.
According to the Pythagorean theorem, we have the equation:
(x^2) + ((x+1)^2) = ((x+2)^2)
Step 4: Solve the equation.
Expanding the equation gives us:
x^2 + (x^2 + 2x + 1) = (x^2 + 4x + 4)
2x^2 + 2x + 1 = x^2 + 4x + 4
Bringing all the terms to one side:
2x^2 + 2x + 1 - x^2 - 4x - 4 = 0
x^2 - 2x - 3 = 0
Factoring the quadratic equation:
(x - 3)(x + 1) = 0
Setting each factor equal to zero:
x - 3 = 0 --> x = 3
x + 1 = 0 --> x = -1
Since we are dealing with lengths, the value of x cannot be negative. Therefore, x = 3 is the only valid solution.
Step 5: Find the length of the smallest side.
Now that we have x = 3, we can substitute it back into our assumption. Thus, the consecutive integers representing the sides of the triangle are: 3, 4, and 5.
Therefore, the length of the smallest side is 3.