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?

Question

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. \( \frac{21}{2} x + \frac{5}{4} \).