The base of triangle is 2 unit less than the sum of the other two sides. The perimeter of the triangle is 48 units and the other two sides differ by 1 unit. Find the sides of triangle.
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The base of triangle is 2 unit less than the sum of the other two sides. The perimeter of the triangle is 48 units and the other two sides differ by 1 unit. Find the sides of triangle.
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Step-by-step explanation:
Let's denote the two sides of the triangle that differ by 1 unit as x and x+1. The base of the triangle will be (x + x + 1) - 2, which simplifies to 2x - 1.
According to the given information, the perimeter of the triangle is 48 units. So, we can write the equation:
Perimeter = x + (x+1) + (2x-1) = 48
Now, let's solve the equation to find the value of x:
3x = 48 + 1
3x = 49
x = 49/3
x ≈ 16.33
Since x represents the length of one side, it cannot have a fractional value in a triangle. However, we know that the other side is x+1. So, we can approximate the values of the sides:
x ≈ 16, and x+1 ≈ 16 + 1 = 17
Now, we can check if the sum of the two sides and the base equals the perimeter:
16 + 17 + (2(16) - 1) = 48
33 + 31 = 48
The sum of the sides and the base indeed equals the perimeter, so the values of the sides of the triangle are approximately 16, 17, and 31 units.
Let's call the length of the base of the triangle "b", and the lengths of the other two sides "x" and "y".
From the problem statement, we know that:
b = x + y - 2 (the base is 2 units less than the sum of the other two sides)
x + y + b = 48 (the perimeter of the triangle is 48 units)
x - y = 1 (the other two sides differ by 1 unit)
We can use substitution to solve for the variables. First, we can substitute the expression for "b" into the equation for the perimeter:
x + y + (x + y - 2) = 48
2x + 2y - 2 = 48
2x + 2y = 50
x + y = 25
Next, we can use the equation x - y = 1 to solve for one of the variables in terms of the other:
y = x - 1
Substituting this into the equation x + y = 25, we get:
x + (x - 1) = 25
2x - 1 = 25
2x = 26
x = 13
Substituting this value back into y = x - 1, we get:
y = 13 - 1 = 12
Finally, we can substitute x and y into the equation for b:
b = x + y - 2
b = 13 + 12 - 2
b = 23
Therefore, the sides of the triangle are 12 units, 13 units, and 23 units.