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

1 mm
1 mm

20 mm
20 mm

5 mm
5 mm

2 mm
2 mm

Answer 1

To determine the possible lengths of the third side of a triangle when two sides are given (10 mm and 7 mm), we can use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides must be greater than the length of the remaining side.

Let the lengths of the sides be \( a = 10 \) mm, \( b = 7 \) mm, and let \( c \) be the length of the third side.

According to the triangle inequality, we have the following conditions:

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

2. \( a + c > b \)
- \( 10 + c > 7 \)
- \( c > 7 - 10 \) (or \( c > -3 \)) [This condition doesn't restrict valid positive values for \( c \)]

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

From these three inequalities, we can determine the range for \( c \):

- From the first inequality: \( c < 17 \)
- From the second inequality: Not restrictive as \( c \) can be any positive value.
- From the third inequality: \( c > 3 \)

Thus, we arrive at the final range for the third side \( c \):

\[
3 < c < 17
\]

Now, let's look at the provided options:

1. **1 mm**: Not valid (since \( 1 < 3 \))
2. **20 mm**: Not valid (since \( 20 > 17 \))
3. **5 mm**: Valid (since \( 3 < 5 < 17 \))
4. **2 mm**: Not valid (since \( 2 < 3 \))

The only valid choice for the length of the third side of the triangle is **5 mm**.