Question

1. find \\(x\\) and the perimeter of the triangle if \\(\delta abc\\) is equilateral.

Explanation:

Equate the side lengths

Since \(\triangle ABC\) is equilateral, all three sides are equal in length. Therefore, we can set the expressions for side \(BC\) and side \(AB\) equal to each other:

$$\frac{x}{6} + 3 = \frac{x}{3} + 1$$

Solve for x

Using the Linear Equation Solving knowledge point:
Multiply the entire equation by the least common denominator, \(6\):

$$x + 18 = 2x + 6$$

Subtract \(x\) from both sides:

$$18 = x + 6$$

Subtract \(6\) from both sides:

$$x = 12$$

Calculate the side length

Substitute \(x = 12\) back into one of the side length expressions:

$$\text{Side length} = \frac{12}{3} + 1 = 4 + 1 = 5$$

Calculate the perimeter

An equilateral triangle has three equal sides. Multiply the side length by \(3\):

$$\text{Perimeter} = 3 \times 5 = 15$$

Answer:

\(x = 12\), Perimeter \(= 15\)