Question

two sides of a triangle measure 6.5 cm and 10cm what would the third side measure

Answer 1

For a triangle with sides measuring \( a \), \( b \), and \( c \), the lengths of any two sides must be greater than the length of the third side. This is known as the triangle inequality theorem.

Given two sides of the triangle are \( 6.5 \, \text{cm} \) and \( 10 \, \text{cm} \), we can denote these sides as \( a = 6.5 \, \text{cm} \) and \( b = 10 \, \text{cm} \). Let \( c \) be the length of the third side.

According to the triangle inequality theorem, the following inequalities must hold:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Applying these inequalities:

1. \( 6.5 + 10 > c \) → \( 16.5 > c \) → \( c < 16.5 \)
2. \( 6.5 + c > 10 \) → \( c > 10 - 6.5 \) → \( c > 3.5 \)
3. \( 10 + c > 6.5 \) → \( c > 6.5 - 10 \) → This is always true since \( c \) must be positive.

Combining the relevant inequalities, we have:

\[
3.5 < c < 16.5
\]

Therefore, the length of the third side \( c \) must be greater than \( 3.5 \, \text{cm} \) and less than \( 16.5 \, \text{cm} \).