**CONNECTING CONCEPTS**  Consider the triangle shown.

**a.** Write a function that represents the perimeter $P$P​ of the triangle.

$P=$P=​  

Question 2

**b.** Identify the independent and dependent variables.

The independent variable is  and the dependent variable is .

Question 3

**c.** Describe the domain and range of the function. (*Hint*: the sum of the lengths of any two sides of a triangle is greater than the length of the remaining side.)

The domain is  $<h<$<h<​ .

The range is  $<P<$<P<​ .

Let's work through the questions one by one.

### Question 1: Write a function for the perimeter $ P $ of the triangle

Assuming the triangle has side lengths depending on $ h $, we can write the perimeter as the sum of the lengths of the three sides.

Let’s denote the three sides of the triangle as:

- Side 1: $ a $
- Side 2: $ b $
- Side 3: $ c(h) $ (this side is expressed as a function of $ h $).

Then, the perimeter $ P $ is the sum of the three sides:

\[
P(h) = a + b + c(h)
\]

You would need to know the specific relationship between $ h $ and the sides to write a fully explicit function. For now, the perimeter function is:

\[
P(h) = a + b + c(h)
\]

### Question 2: Identify the independent and dependent variables

- The **independent variable** is $ h $, as it controls the shape or size of the triangle (possibly a height).
- The **dependent variable** is $ P $, as the perimeter depends on $ h $.

### Question 3: Describe the domain and range of the function

#### Domain:
The domain refers to all possible values of $ h $ for which the triangle inequality holds. The triangle inequality states that the sum of the lengths of any two sides must be greater than the remaining side.

Let the domain be expressed as:

\[
\text{Domain: } d_1 < h < d_2
\]

Where $ d_1 $ and $ d_2 $ are the minimum and maximum bounds of $ h $ such that a valid triangle is formed.

#### Range:
The range describes the possible values of the perimeter $ P $. Since the perimeter is a function of $ h $, it will have a minimum and maximum value based on the valid $ h $-values.

Let the range be expressed as:

\[
\text{Range: } P_{\text{min}} < P < P_{\text{max}}
\]

Again, the specific values would depend on how $ h $ affects the lengths of the sides of the triangle.

---

Would you like further clarifications on any part of this solution? Here are five related questions to deepen your understanding:

1. How would you write the function for $ c(h) $ if the triangle's sides depend on geometry?
2. What happens to the perimeter function if one side remains constant while $ h $ changes?
3. How would you calculate specific values for the perimeter given certain values of $ h $?
4. Could there be a case where the perimeter has a maximum value? If so, under what conditions?
5. How does the triangle inequality affect the limits on the domain of $ h $?

**Tip**: Always check the triangle inequality when working with geometric problems involving variable side lengths to ensure valid triangles are formed.

CONNECTING CONCEPTS  Consider the triangle shown.

a. Write a function that represents the perimeter P of the triangle.

b. Identify the independent and dependent variables.

c. Describe the domain and range of the function.

Explore how to write a function for the perimeter of a triangle, identify the independent and dependent variables, and determine the domain and range of the function. Learn key concepts in geometry, functions, and the triangle inequality, suitable for students in Grades 8-10.

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