The perimeter of an isosceles triangle is 28 one of is sides is (6y-2x +1),another is 4y then the base is x+2y.find x and y hence state the lengths of the sides of the triangle
If x+ 2y is the base , then 6y - 2x +1 = 4y
2y - 2x = -1
and x+2y + 4y + 6y -2x + 1= 28
12y - x = 27 or x = 12y - 27
Use substitution to find x and y
To find the lengths of the sides of the triangle, we need to use the given information about the perimeter and the sides:
1. Perimeter of the triangle: 28
2. One of the sides: 6y - 2x + 1
3. Another side: 4y
4. Base: x + 2y
We can set up an equation using the given information:
Perimeter = Side 1 + Side 2 + Base
28 = (6y - 2x + 1) + 4y + (x + 2y)
Simplifying the right side of the equation:
28 = 6y - 2x + 1 + 4y + x + 2y
Combine like terms:
28 = 12y - x + 1 + x + 2y
Combine like terms:
28 = 14y + 1
Subtract 1 from both sides:
27 = 14y
Divide both sides by 14:
y = 27/14
Thus, y is equal to 27/14.
Next, we can substitute the value of y back into the equation to find x.
28 = (6y - 2x + 1) + 4y + (x + 2y)
Replacing y with 27/14:
28 = (6(27/14) - 2x + 1) + 4(27/14) + (x + 2(27/14))
Simplifying:
28 = (27/7 - 2x + 1) + (54/14 + x + 27/7)
28 = (27/7 - 2x + 1) + (27/7 + x + 27/7)
Combine like terms:
28 = (54/7 - 2x) + (54/7 + x)
28 = (54 - 14x)/7 + (54 + 7x)/7
Combine the fractions:
28 = (108 - 14x + 54 + 7x)/7
Multiply both sides by 7 to remove the denominator:
196 = 108 - 14x + 54 + 7x
Combine like terms:
196 = 162 - 7x
Subtract 162 from both sides:
34 = -7x
Divide both sides by -7:
x = -34/-7
x = 34/7
Thus, x is equal to 34/7.
Now that we have found the values of x and y, we can substitute them back into the expressions for the sides to find their lengths:
Side 1: 6y - 2x + 1
Substituting y = 27/14 and x = 34/7:
Side 1 = 6(27/14) - 2(34/7) + 1
= 81/7 - 68/7 + 1
= 14/7
= 2
Side 2: 4y
Substituting y = 27/14:
Side 2 = 4(27/14)
= 27/7
= 3
Base: x + 2y
Substituting x = 34/7 and y = 27/14:
Base = (34/7) + 2(27/14)
= 34/7 + 27/7
= 61/7
Therefore, the lengths of the sides of the triangle are:
Side 1: 2
Side 2: 3
Base: 61/7