Determining the Possible Lengths of the Third Side in a Triangle

When two sides of a triangle are known, the triangle inequality theorem can be used to determine the possible lengths of the third side. 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's explore how this applies in a specific scenario where the two sides are known to be 3 and 7.

Triangle Inequality Theorem

Given two sides of a triangle, a and b, with lengths 3 and 7 respectively, the third side, c, must satisfy the following inequalities:

a b > c a c > b b c > a

Substituting the values of a and b, we get:

3 7 > c → c 3 c > 7 → c > 4 7 c > 3 → c > -4 (this is always true since c must be positive)

Combining the inequalities, we find:

4

Therefore, the possible lengths for the third side c must be greater than 4 and less than 10.

Special Cases and the Law of Cosines

For a more complex scenario, let's consider the case where the sides are 3, 8, and an unknown third side, c. The triangle inequality theorem still applies:

3 8 > c → c 3 c > 8 → c > 5 8 c > 3 → c > -5 (always true since c must be positive)

Thus, combining the inequalities, we get:

5

If the angle between the sides with lengths 3 and 8 is known, the law of cosines can be used to find the length of the third side. The law of cosines states:

c^2 a^2 b^2 - 2ab*cos(C)

Substituting a 3 and b 8, we get:

c^2 3^2 8^2 - 2*3*8*cos(C) 9 64 - 48*cos(C)

c^2 73 - 48*cos(C)

When C is 60 degrees, cos(C) 1/2, so:

c sqrt(73 - 48*0.5) sqrt(73 - 24) sqrt(49) 7

In summary, if the sides of the triangle are 3 and 8, the possible lengths for the third side, c, are any values between 5 and 11, inclusive. This range is a direct result of the triangle inequality theorem, ensuring that the sides can form a valid triangle.