To determine the perimeter of triangle $\triangle DEF$ with vertices at $D(1,3)$, $E(-3,2)$, and $F(-2,-2)$, we need to calculate the lengths of the three sides using the distance formula:

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step 1: Calculate the distance between $D$ and $E$ ($DE$):

$DE = \sqrt{((-3) - 1)^2 + (2 - 3)^2} = \sqrt{(-4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.1$

Step 2: Calculate the distance between $E$ and $F$ ($EF$):

$EF = \sqrt{((-2) - (-3))^2 + (-2 - 2)^2} = \sqrt{(1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \approx 4.1$

Step 3: Calculate the distance between $D$ and $F$ ($DF$):

$DF = \sqrt{((-2) - 1)^2 + (-2 - 3)^2} = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.8$

Step 4: Calculate the perimeter of the triangle

The perimeter $P$ is the sum of the lengths of the sides:

$P = DE + EF + DF \approx 4.1 + 4.1 + 5.8 = 14.0$

Answer:

The perimeter of triangle $DEF$ is approximately 14.0, which rounds to 14.1.

Therefore, the correct answer is:

b. 14.1

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Would you like further details or explanations? Here are some related questions to explore:

1. How do you use the distance formula in geometry problems?
2. What is the formula for calculating the area of a triangle using its vertices?
3. Can you explain how to classify a triangle based on its side lengths?
4. How would you calculate the perimeter of a triangle using only its angles?
5. How do you find the midpoint of a line segment between two points?

Tip: Remember to always check your work with a calculator for precision when dealing with decimal approximations!