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

Question

Responses

5 mm
5 mm

2 mm
2 mm

20 mm

1 mm

Answer 1

To determine the length of the third side of a triangle when two sides are given, you 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 remaining side.

Given two sides of the triangle:

1. \( a = 10 , \text{mm} \)
2. \( b = 7 , \text{mm} \)

Let \( c \) be the length of the third side. The triangle inequality gives us the following conditions:

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

Substituting the values:

1. \( 10 + 7 > c \)

   • \( 17 > c \)
   • Thus, \( c < 17 \)

2. \( 10 + c > 7 \)

   • \( c > 7 - 10 \)
   • \( c > -3 \) (this condition is always satisfied since side lengths are positive)

3. \( 7 + c > 10 \)

   • \( c > 10 - 7 \)
   • \( c > 3 \)

Combining the valid inequalities, we have:

• \( 3 < c < 17 \)

Now let's evaluate the choices given:

1. 5 mm (This is valid, as it's between 3 and 17)
2. 2 mm (Invalid, as it's not greater than 3)
3. 20 mm (Invalid, as it's not less than 17)
4. 1 mm (Invalid, as it's not greater than 3)

The only valid length for the third side from the provided options is: 5 mm.