Question

Two sides of a triangle are 10 mm and 7 mm. Determine the length of the third side from the choices given.(1 point)
Responses

2 mm
2 mm

1 mm
1 mm

20 mm
20 mm

5 mm

Answer 1

To determine the possible length of the third side of a triangle when the lengths of the other two sides are known (10 mm and 7 mm), we can apply 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 the third side be denoted as \( x \). According to the triangle inequality, we have the following conditions:

1. \( 10 + 7 > x \) (which simplifies to \( x < 17 \))
2. \( 10 + x > 7 \) (which simplifies to \( x > -3 \), not useful since sides must be positive)
3. \( 7 + x > 10 \) (which simplifies to \( x > 3 \))

From the inequalities above, we find the valid range for \( x \):

- From the first inequality: \( x < 17 \)
- From the third inequality: \( x > 3 \)

Thus, combining these conditions, we get:
\[ 3 < x < 17 \]

Now we can evaluate the given choices:

1. **2 mm**: This does not satisfy \( x > 3 \).
2. **1 mm**: This does not satisfy \( x > 3 \).
3. **20 mm**: This does not satisfy \( x < 17 \).
4. **5 mm**: This satisfies \( 3 < x < 17 \).

The only choice that fits the criteria is **5 mm**.

So, the length of the third side can only be **5 mm**.