Questions: Write a simplified expression that represents the perimeter of an irregular quadrilateral with side lengths (2 1/4 t-5), (4 t+3), (1/2 t-1), and (3 t+2).

Transcript text: Model Problems Write a simplified expression that represents the perimeter of an irregular quadrilateral with side lengths $\left(2 \frac{1}{4} t-5\right),(4 t+3),\left(\frac{1}{2} t-1\right)$, and $(3 t+2)$.

Solution

Solution Steps

Step 1: Define the Side Lengths

The side lengths of the irregular quadrilateral are given as:

$ \text{side1} = 2 \frac{1}{4} t - 5 = \frac{9}{4} t - 5 $
$ \text{side2} = 4t + 3 $
$ \text{side3} = \frac{1}{2} t - 1 $
$ \text{side4} = 3t + 2 $

Step 2: Sum the Side Lengths

To find the perimeter $ P $, we sum all the side lengths: \[ P = \left( \frac{9}{4} t - 5 \right) + (4t + 3) + \left( \frac{1}{2} t - 1 \right) + (3t + 2) \]

Step 3: Simplify the Expression

Combining like terms, we simplify the expression for the perimeter: \[ P = \left( \frac{9}{4} t + 4t + \frac{1}{2} t + 3t \right) + (-5 + 3 - 1 + 2) \] This results in: \[ P = \frac{39}{4} t - 1 \] Thus, the simplified expression for the perimeter of the irregular quadrilateral is: \[ P = \frac{39}{4} t - 1 \]

Final Answer

$\boxed{P = \frac{39}{4} t - 1}$

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