A triangle has side lengths of left parenthesis, 2, b, plus, 5, c, right parenthesis(2b+5c) centimeters, left parenthesis, 10, b, plus, 7, d, right parenthesis(10b+7d) centimeters, and left parenthesis, 4, d, minus, 3, c, right parenthesis(4d−3c) centimeters. Which expression represents the perimeter, in centimeters, of the triangle?
Answer

To find the perimeter of a triangle, you simply sum the lengths of all three sides.

The sides of the triangle are given as:

1. \( (2b + 5c) \) cm
2. \( (10b + 7d) \) cm
3. \( (4d - 3c) \) cm

The perimeter \( P \) of the triangle is the sum of these three expressions:

\[
P = (2b + 5c) + (10b + 7d) + (4d - 3c)
\]

Now, we combine like terms:

- For \( b \)-terms: \( 2b + 10b = 12b \)
- For \( c \)-terms: \( 5c - 3c = 2c \)
- For \( d \)-terms: \( 7d + 4d = 11d \)

Thus, the expression for the perimeter is:

\[
P = 12b + 2c + 11d
\]

Therefore, the perimeter of the triangle is:

\[
\boxed{12b + 2c + 11d \text{ cm}}
\]

Would you like further details or have any questions about the solution?

Here are 5 related questions to explore:
1. How do you calculate the perimeter of a polygon with more sides?
2. What is the perimeter of a triangle with specific values for \( b \), \( c \), and \( d \)?
3. How do you solve for the area of a triangle given side lengths?
4. What is the relationship between perimeter and area in a triangle?
5. Can you express the perimeter of a triangle in terms of another variable?

**Tip**: Always double-check your work by simplifying algebraic expressions step by step to avoid mistakes.

A triangle has side lengths of (2b+5c) cm, (10b+7d) cm, and (4d−3c) cm. Which expression represents the perimeter, in centimeters, of the triangle?

Learn how to calculate the perimeter of a triangle with sides expressed in algebraic terms, such as (2b + 5c), (10b + 7d), and (4d - 3c). This problem covers key algebra concepts, including simplifying expressions and perimeter calculations, and is suitable for students in Grades 8-10.

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