If the two sides of a triangle are 50 and 12, the range of the third side is between  and  .

In a triangle, the length of any side must satisfy the **triangle inequality theorem**, which states:

1. The sum of any two sides must be greater than the third side.
   
Given two sides \( a = 50 \) and \( b = 12 \), the length of the third side, \( c \), must satisfy the following conditions:
   
\[
a + b > c
\]
\[
a + c > b
\]
\[
b + c > a
\]

Let's solve for \( c \):

1. \( 50 + 12 > c \implies c < 62 \)
2. \( 50 + c > 12 \implies c > 38 \)

Thus, the length of the third side \( c \) must satisfy:

\[
38 < c < 62
\]

So, the range of the third side is between **38 and 62**.

Would you like more details or have any questions?

Here are five related questions to expand on this:

1. What happens if the third side is exactly 38 or 62?
2. How can we calculate the area of a triangle if we know all three sides?
3. How do you calculate angles given the sides of a triangle?
4. What is the relationship between the perimeter and the sides of a triangle?
5. How does the triangle inequality theorem apply to obtuse or acute triangles?

**Tip:** When applying the triangle inequality theorem, always check each side combination to ensure the triangle can exist!

If the two sides of a triangle are 50 and 12, the range of the third side is between  and .

Learn how to apply the triangle inequality theorem to find the range of the third side of a triangle with sides 50 and 12. Using the conditions a + b > c, a + c > b, and b + c > a, we find the third side must be between 38 and 62. Suitable for students in Grades 8-10.

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