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 2 mm 2 mm 20 mm 20 mm 5 mm

Answer 1

To find the possible lengths of the third side of a triangle when two sides are given, 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 the lengths of the two given sides be \( a = 10 \) mm and \( b = 7 \) mm. We can denote the length of the third side as \( c \).

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

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

2. \( a + c > b \)
\( 10 + c > 7 \)
\( c > 7 - 10 \)
\( c > -3 \) (which is always true for positive \( c \))

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

Combining these inequalities, we find that the length of the third side \( c \) must satisfy:

\( 3 < c < 17 \)

Now, let's evaluate the response options:
- 1 mm: does not satisfy \( c > 3 \)
- 2 mm: does not satisfy \( c > 3 \)
- 20 mm: does not satisfy \( c < 17 \)
- 5 mm: satisfies \( 3 < c < 17 \)

Thus, the only valid choice for the length of the third side from the given options is **5 mm**.