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

A. 21/2 x

B. 5/4 x -3

C. 47/4 x

D. 21/2 x +5/4

Answer 1

To find the perimeter of the triangle represented by the given polynomial side lengths, we need to add the lengths of the three sides together.

The side lengths are:
1. \( \frac{3}{2}x + \frac{5}{2} \)
2. \( 5x - \frac{3}{2} \)
3. \( 4x + \frac{1}{4} \)

Now, let's add these expressions together step-by-step.

### Step 1: Combine the polynomial terms involving \(x\)
1. The first term is \( \frac{3}{2}x \).
2. The second term is \( 5x = \frac{10}{2}x \).
3. The third term is \( 4x = \frac{8}{2}x \).

Adding these together:
\[
\frac{3}{2}x + \frac{10}{2}x + \frac{8}{2}x = \left(\frac{3 + 10 + 8}{2}\right)x = \frac{21}{2}x
\]

### Step 2: Combine the constant terms
1. The first constant term is \( \frac{5}{2} \).
2. The second constant term is \( -\frac{3}{2} \).
3. The third constant term is \( \frac{1}{4} \).

We start by combining the first two constants:
\[
\frac{5}{2} - \frac{3}{2} = \frac{2}{2} = 1
\]

Now, we need to add \( 1 \) (which can be expressed as \( \frac{4}{4} \)) to \( \frac{1}{4} \):
\[
1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}
\]

### Step 3: Combine both results to find the perimeter
Now, we add the results of the polynomial terms and the constant terms:
\[
\text{Perimeter} = \frac{21}{2}x + \frac{5}{4}
\]

Thus, the expression that represents the perimeter of the triangle is:
\[
\frac{21}{2}x + \frac{5}{4}
\]

The correct option is: **D. \( \frac{21}{2} x + \frac{5}{4} \)**.