Question

Find the range for the measure of the third side of a triangle given the measures of two sides.

a. 18 ft and 23 ft
b. 6 ft ; n ; 40 ft
c. 17 ft; n ; 24 ft
d. 5 ft; n ; 41 ft
e. 18 ft; n; 23 ft

Answer 1

The range for the third side of a triangle follows from the triangle inequality theorem, which states that the length of any side must be greater than the difference and less than the sum of the other two sides.

Step-by-step explanation:

To find the range for the measure of the third side of a triangle given the measures of two sides, one 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 remaining side.

1. For the two sides measuring 18 ft and 23 ft, the third side, n, must be greater than the difference and less than the sum of these two side lengths. Therefore, 5 ft < n < 41 ft.
2. For the two sides measuring 6 ft and 40 ft, similar logic applies, giving us n > 34 ft and n < 46 ft.
3. With sides measuring 17 ft and 24 ft, the range for the third side would be 7 ft < n < 41 ft.
4. For sides measuring 5 ft and 41 ft, the range for the third side is 36 ft < n < 46 ft.
5. Lastly, if two sides are both 18 ft and 23 ft again, the range doesn't change, thus it remains 5 ft < n < 41 ft.

To clarify further with an example, if you have two sides of a triangle with lengths 'a' and 'b', the length of the third side 'c' must satisfy the two inequalities: a + b > c and |a - b| < c.