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

1 mm
1 mm

20 mm
20 mm

2 mm

Answer 1

To determine the lengths for the third side of a triangle with two sides measuring 10 mm and 7 mm, you can use the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.

Let's denote the sides as follows:

• Side A = 10 mm
• Side B = 7 mm
• Side C = length of the third side (unknown)

According to the triangle inequality theorem, we have the following conditions that must be met:

1. \( A + B > C \)
2. \( A + C > B \)
3. \( B + C > A \)

Substituting in the values we know:

1. \( 10 + 7 > C \) simplifies to \( C < 17 \)
2. \( 10 + C > 7 \) simplifies to \( C > -3 \) (this condition is always satisfied since lengths are positive)
3. \( 7 + C > 10 \) simplifies to \( C > 3 \)

From the valid inequalities, we find:

• \( C < 17 \)
• \( C > 3 \)

Thus, the length of the third side must be in the range \( 3 < C < 17 \).

Now, evaluating the choices:

• 5 mm: valid (between 3 and 17)
• 1 mm: not valid (less than 3)
• 20 mm: not valid (greater than 17)
• 2 mm: not valid (less than 3)

The only valid option is 5 mm.