If the perimeter of the triangle is 22 cm find the length of the shortest side the first side is 3x+1 the second side is 2x-1 the third side is x+4

We are given that the perimeter of the triangle is 22 cm, and the expressions for the three sides of the triangle are:

- First side: \( 3x + 1 \)
- Second side: \( 2x - 1 \)
- Third side: \( x + 4 \)

### Step 1: Use the formula for the perimeter
The perimeter of a triangle is the sum of the lengths of all its sides. Therefore, we can write:

\[
(3x + 1) + (2x - 1) + (x + 4) = 22
\]

### Step 2: Simplify the equation
Combine the like terms:

\[
3x + 2x + x + 1 - 1 + 4 = 22
\]

This simplifies to:

\[
6x + 4 = 22
\]

### Step 3: Solve for \(x\)
Subtract 4 from both sides:

\[
6x = 18
\]

Now, divide by 6:

\[
x = 3
\]

### Step 4: Find the lengths of the sides
Now that we know \(x = 3\), substitute this value into the expressions for the sides:

- First side: \( 3(3) + 1 = 9 + 1 = 10 \)
- Second side: \( 2(3) - 1 = 6 - 1 = 5 \)
- Third side: \( 3 + 4 = 7 \)

### Step 5: Identify the shortest side
The three sides of the triangle are 10 cm, 5 cm, and 7 cm. Therefore, the shortest side is **5 cm**.

Would you like more details or explanations? 

Here are 5 related questions you can explore:
1. How do you find the perimeter of a triangle when given the side lengths?
2. Can a triangle with sides 10 cm, 5 cm, and 7 cm exist? (Hint: triangle inequality)
3. How would the problem change if the perimeter was 30 cm instead of 22 cm?
4. What if the expressions for the sides were in terms of different variables?
5. Can you generalize this process for any polygon?

**Tip:** Always check the triangle inequality to ensure the side lengths form a valid triangle!

If the perimeter of the triangle is 22 cm, find the length of the shortest side. The first side is 3x+1, the second side is 2x-1, the third side is x+4.

This detailed step-by-step solution shows how to find the length of the shortest side of a triangle given the perimeter of 22 cm. The sides are represented by algebraic expressions, and the problem covers key concepts like solving linear equations and applying the perimeter formula. Ideal for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation