# What are the possible lengths of the third side of the triangle?

I have this for a homework question: "The side lengths of a triangle are $11.3$ centimeters, $14.7$ centimeters, and $x$ centimeters. The perimeter of the triangle is less than $44$ centimeters. What are the possible values of $x$?"

No idea what to do.. I can do this type of math on other problems, but THIS one, I'm not sure about. It would be nice if someone could help me out and show me the answer and how it was gotten!

• HINT: With the two sides given (ignore the 44 for the moment) what is the longest third side you can have, and what is the shortest? How does the constraint on the perimeter affect that? – Mark Bennet Oct 12 '15 at 20:40
• You need to use the Triangle Inequality. – N. F. Taussig Oct 12 '15 at 20:44
• Actually you don't use the triangle inequality in the end as the perimeter being less than 44 is a stronger restriction. You have to check the triangle inequality to assure that it isn't the stronger restriction, though. – fleablood Oct 12 '15 at 21:25

## 3 Answers

Your perimeter is the total of all three sides. You know two sides. Add them up. What is the maximum length that the third side can be without exceeding the perimeter of $44$?

If you really want to visualize this, create an angle with two straws, labeling them with the lengths the represent.

How narrow can you make the angle between them? What is the closest the two opposite ends can get? That's your minimum length for the third side.

• You are forgetting the third restriction. Given the lengths of the two sides how far apart can the endpoints of the two sides b?. In this case that doesn't matter as the perimeter restriction makes the triangle inequality restriction redundant but in general it needs to be considered. – fleablood Oct 13 '15 at 15:49
• @fleablood Good point. I made an unfounded assumption. – Adam Hrankowski Oct 13 '15 at 15:52
• Well, in my first answer I totally forgot about the minimum value... – fleablood Oct 13 '15 at 16:08

The shortest distance between two points is a straight line, right? So one side of a triangle is a straight line. The path going along the other to sides isn't a straight line. So the side of a triangle is shorter than the sum of the other two sides.

So if a triangle has sides a, b, and c. c < a + b. a < b + c . And b < a + c. Always. (if they are proper triangles.)

Also if you have a triangle where one side is longer than the other, say c > b, then, as a + b > c, then a > c - b. Or in general c > |b - a|, b > |c -b| and a > |c - b|. This makes sense as is the distance of a third leg of a triangle has to be longer than simply going up one distant of one leg and straight back down the leg another.

Call the unknown side of the triangle x. So here you have three pieces of information:

Perimeter = 11.3 + 14.7 + x <= 44, Triangle Inequality: x < 11.3 + 14.7 ,Triangle Inequality variation: x > 14.7 - 11.3

The first one $\rightarrow$ 26 + x <= 44 $\rightarrow$ x <= 18.

The second one $\rightarrow$ x < 26. (This is weaker than the first one so we don't need it; if x <=18 then it's also x < 26).

The third one $\rightarrow$ x > 3.4 $\rightarrow$ 3.4 < x.

Combine to get 3.4 < x <= 18.

Any real number between 3.4 and 18 is an answer.

• Don't forget about the lower bound. – Tebbe Oct 13 '15 at 11:56
• Oh, I did. Didn't I? – fleablood Oct 13 '15 at 15:28

If perimeter 44 - 11.3 - 14.7 = x, upper bound is 18.

The lower bound in place of 44 occurs when the sides are in a straight line making x = 14.7 - 11.3 = 3.4