Given are the lengths of two sides of a triangle. Find the range of lengths for the third side.

Question

1.) 10, 13

2.) 5, 12

3.) 11, 4

4.) 9, 7

5.) 4, 3

6.)2, 7

Answer 1

The triangle inequality states that in a triangle, any two of the side lengths must add up to a length that is greater than the third. That is, if you have a triangle with sides a, b, and c, then a+b>c, a+c>b, and b+c>a.

In this case, we are given two lengths of a triangle. Let's call those two values a and b, where a>b.

Then we have that c<a+b, right off the bat. This sets an upper limit of the third side length as the sum of the side lengths.

Now we can also use another inequality, a<b+c. By rearranging the inequality, we get that a-b<c, giving us a lower limit for the third side length of the difference in side lengths.

Now we can apply these limits to all the individual questions.

1) 13 - 10 < c < 13 + 10. This simplifies down to 3 < c < 23

2) 12 - 5 < c < 12 + 5. This simplifies down to 7 < c < 17

3) 11 - 4 < c < 11 + 4. This simplifies down to 7 < c < 15

It should be easy to apply this to the remaining questions. Please reply if you need additional help, or if something is unclear.