Question

A triangle has sides whose lengths in units are represented by polynomials as shown, where `x` is a positive integer. Which expression represents the perimeter of the triangle in units?

the units to the traingle are 3/2 x + 5/2, 5x-3/2, 4x+1/4


21/2 x

5/4 x -3

47/4 x

21/2 x +5/4

Answer 1

To find the perimeter of the triangle given the lengths of its sides as polynomials, we need to sum up the lengths of all three sides:

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

Let's add these expressions together step by step.

First, we can write down the sum of the sides:

\[
\text{Perimeter} = \left(\frac{3}{2}x + \frac{5}{2}\right) + \left(5x - \frac{3}{2}\right) + \left(4x + \frac{1}{4}\right)
\]

Now, we will combine the coefficients of \(x\) and the constant terms:

1. Coefficients of \(x\):
- From the first side: \( \frac{3}{2}x \)
- From the second side: \( 5x \)
- From the third side: \( 4x \)

Summing these:

\[
\frac{3}{2} + 5 + 4 = \frac{3}{2} + \frac{10}{2} + \frac{8}{2} = \frac{21}{2}x
\]

2. Constant terms:
- From the first side: \( \frac{5}{2} \)
- From the second side: \( -\frac{3}{2} \)
- From the third side: \( \frac{1}{4} \)

Summing these:

\[
\frac{5}{2} - \frac{3}{2} + \frac{1}{4} = \frac{2}{2} + \frac{1}{4} = 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}
\]

Now, combining both parts gives us the total perimeter:

\[
\text{Perimeter} = \frac{21}{2}x + \frac{5}{4}
\]

Therefore, the expression that represents the perimeter of the triangle in units is:

\[
\boxed{\frac{21}{2}x + \frac{5}{4}}
\]