Question

two sides of a triangle are 10mm and 7mm, determine the length of the third side

Answer 1

To determine the possible lengths of the third side of a triangle with two sides measuring 10 mm and 7 mm, we can use the triangle inequality theorem. This 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 \( a = 10 \) mm, \( b = 7 \) mm, and \( c \) be the length of the third side. According to the triangle inequality theorem, we can establish the following inequalities:

1. \( a + b > c \)
\( 10 + 7 > c \)
\( 17 > c \)
\( c < 17 \)

2. \( a + c > b \)
\( 10 + c > 7 \)
\( c > 7 - 10 \) (which simplifies to \( c > -3 \), but since side lengths must be positive)
\( c > 0 \)

3. \( b + c > a \)
\( 7 + c > 10 \)
\( c > 10 - 7 \)
\( c > 3 \)

Combining the inequalities, we have:

\[
3 < c < 17
\]

So the length of the third side \( c \) must be greater than 3 mm and less than 17 mm.