The sides of an equilateral triangle are 3a+2, 3b-a and b+3 respectively. what is the value of a and b and find the perimeter of the triangle
All sides of an equilateral triangle are equal.
So,
3a + 2 = 3b - a
=> 4a - 3b = -2 ....1
3b - a = b + 3
=> 2b - 3 = a ....2
Put (2b-3=a) in ....1
=> 4(2b - 3) - 3b = -2
=> 8b - 12 - 3b = -2
=> 5b = 10
=> b = 2
Now you can obtain 'a' too
Add all sides to find the perimeter.
To find the values of a and b, we can equate the given side lengths to each other.
We have the following equations:
3a + 2 = 3b - a (Equation 1)
3b - a = b + 3 (Equation 2)
We can solve this system of equations to find the values of a and b.
Let's solve Equation 1 for a:
3a + 2 = 3b - a
4a = 3b - 2
a = (3b - 2) / 4 (Equation 3)
Now, substitute this value of a in Equation 2:
3b - (3b - 2) / 4 = b + 3
12b - 3b + 2 = 4b + 12
9b + 2 = 4b + 12
9b - 4b = 12 - 2
5b = 10
b = 2 (Equation 4)
Now we can substitute the value of b in Equation 3 to find a:
a = (3(2) - 2) / 4
a = (6 - 2) / 4
a = 4 / 4
a = 1
So, a = 1 and b = 2.
To find the perimeter of an equilateral triangle, we can use the formula P = 3s, where P is the perimeter and s is the length of each side.
The length of each side in this case is given as 3a + 2, 3b - a, and b + 3.
Substituting the values of a and b:
Side 1 = 3(1) + 2 = 5
Side 2 = 3(2) - 1 = 5
Side 3 = 2 + 3 = 5
Since all three sides are equal, the perimeter is:
P = 3(5) = 15
Therefore, the value of a is 1, the value of b is 2, and the perimeter of the triangle is 15.
To find the values of a and b, we can equate the three sides of the equilateral triangle.
Given the sides of the triangle are:
Side 1: 3a + 2
Side 2: 3b - a
Side 3: b + 3
Since it is an equilateral triangle, all three sides are equal. We can set up equations to solve for a and b.
Equating Side 1 and Side 2:
3a + 2 = 3b - a
Simplifying the equation, we get:
4a = 3b - 2 -- (Equation 1)
Equating Side 2 and Side 3:
3b - a = b + 3
Simplifying the equation, we get:
2b = a + 3 -- (Equation 2)
Now we have two equations. We can solve them simultaneously to find the values of a and b.
First, let's solve Equation 2 for a:
a = 2b - 3
Substituting this value of a in Equation 1:
4(2b - 3) = 3b - 2
Simplifying the equation:
8b - 12 = 3b - 2
5b = 10
b = 2
Now substitute the value of b in Equation 2:
2(2) = a + 3
4 = a + 3
a = 1
Therefore, the values of a and b are:
a = 1
b = 2
To find the perimeter of the triangle, we need to find the sum of all three sides.
Perimeter = Side 1 + Side 2 + Side 3
Perimeter = (3a + 2) + (3b - a) + (b + 3)
Substituting the values of a and b, we get:
Perimeter = (3(1) + 2) + (3(2) - 1) + (2 + 3)
Perimeter = 5 + 5 + 5
Perimeter = 15
Therefore, the perimeter of the equilateral triangle is 15.