Two sides of a triangles are 12 cm and 7 cm , find the range for the length of its third side.

Answer

To find the range for the length of the third side of a triangle when two sides are given, we can use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's denote the lengths of the two given sides as:
- Side 1 = 12 cm
- Side 2 = 7 cm
- Side 3 = x cm (the length we want to find)

### Step 1: Apply the Triangle Inequality Theorem

According to the triangle inequality theorem, we have the following inequalities:

1. \( x + 7 > 12 \)
2. \( x + 12 > 7 \)
3. \( 12 + 7 > x \)

### Step 2: Solve Each Inequality

**Inequality 1:**
\[ x + 7 > 12 \]
Subtract 7 from both sides:
\[ x > 12 - 7 \]
\[ x > 5 \]

**Inequality 2:**
\[ x + 12 > 7 \]
Subtract 12 from both sides:
\[ x > 7 - 12 \]
\[ x > -5 \]
(This inequality does not provide any new information since x must be positive.)

**Inequality 3:**
\[ 12 + 7 > x \]
\[ 19 > x \]
or
\[ x < 19 \]

### Step 3: Combine the Results

From the inequalities we solved:
- \( x > 5 \)
- \( x < 19 \)

We can combine these results to find the range for the length of the third side:
\[ 5 < x < 19 \]

### Final Answer:
The length of the third side \( x \) must be greater than 5 cm and less than 19 cm.

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