Question

Find the perimeter of the figure if the sides are 2x-1, 1/3x, 1/6x-1, 2x-2, x+4, and 3x+4.

Answer 1

To find the perimeter of the figure with the given side lengths, we need to sum up all the side lengths. The side lengths are:

1. \(2x - 1\)
2. \(\frac{1}{3}x\)
3. \(\frac{1}{6}x - 1\)
4. \(2x - 2\)
5. \(x + 4\)
6. \(3x + 4\)

The perimeter \(P\) can be expressed as:

\[
P = (2x - 1) + \left(\frac{1}{3}x\right) + \left(\frac{1}{6}x - 1\right) + (2x - 2) + (x + 4) + (3x + 4)
\]

Now, let’s combine these terms step by step.

1. Combine all the coefficients of \(x\):

\[
2x + \frac{1}{3}x + \frac{1}{6}x + 2x + x + 3x
\]

To add these, first find a common denominator for the fractions. The common denominator among \(1\), \(3\), and \(6\) is \(6\).

Convert each term:

- \(2x = \frac{12}{6}x\)
- \(\frac{1}{3}x = \frac{2}{6}x\)
- \(\frac{1}{6}x\) remains the same.
- \(2x = \frac{12}{6}x\)
- \(x = \frac{6}{6}x\)
- \(3x = \frac{18}{6}x\)

Now we can sum these values:

\[
\frac{12}{6}x + \frac{2}{6}x + \frac{1}{6}x + \frac{12}{6}x + \frac{6}{6}x + \frac{18}{6}x = \frac{51}{6}x
\]

2. Now, combine the constant terms:

\[
-1 - 1 - 2 + 4 + 4 = -2 + 8 = 6
\]

Now, we can put all of this together to find the perimeter:

\[
P = \frac{51}{6}x + 6
\]

Thus, the perimeter of the figure is:

\[
\boxed{\frac{51}{6}x + 6}
\]