if the three sides of an equilateral triangle are 3x + 2, 2y - X and y + 3. find X and y and the perimeter of the triangle. kindly elucidate further on this question
equilateral ---> all sides are equal, so
3x+2 = 2y-x
4x + 2 = 2y
2x + 1 = y
also 3x+2 = y + 3
3x+2 = 2x+1 + 3
x = 2
then y = 5
so each side is 3x+2 = 8
and the perimeter is 24
Given 3 sides of the equilateral triangle are x+y+2, 2x-y-2 and x-y. Find the perimeter
To find the values of X and y and the perimeter of the equilateral triangle, we need to set up equations based on the given information.
Since it is an equilateral triangle, all three sides are equal. So, we can equate the expressions for the sides and solve for X and y.
Equating the expressions for the sides:
3x + 2 = 2y - x = y + 3
We have three equations:
1) 3x + 2 = 2y - x
2) 3x + 2 = y + 3
3) 2y - x = y + 3
Now, let's solve them step by step:
1) 3x + 2 = 2y - x
Combining like terms:
4x + 2 = 2y
Rearranging:
4x = 2y - 2
2x = y - 1
x = (y - 1)/2
2) 3x + 2 = y + 3
3x = y + 1
x = (y + 1)/3
Now that we have obtained expressions for X and y, we can substitute them back into any of the original equations to find their values.
Let's substitute x = (y - 1)/2 into equation 3:
2y - (y - 1)/2 = y + 3
Multiplying through by 2 to eliminate the fraction:
4y - (y - 1) = 2y + 6
4y - y + 1 = 2y + 6
3y + 1 = 2y + 6
3y - 2y = 6 - 1
y = 5
Now, let's substitute y = 5 into x = (y + 1)/3:
x = (5 + 1)/3
x = 2
We have found the values of X and y.
To find the perimeter of the triangle, we add up the lengths of all three sides since it is an equilateral triangle.
Perimeter = (3x + 2) + (2y - x) + (y + 3)
Perimeter = 3x - x + 2y + 2 + y + 3
Perimeter = 2x + 3y + 5
Substituting x = 2 and y = 5 into the equation:
Perimeter = 2(2) + 3(5) + 5
Perimeter = 4 + 15 + 5
Perimeter = 24
Therefore, X = 2, y = 5, and the perimeter of the equilateral triangle is 24 units.