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Question Video: Identifying the Possible Lengths of the Third Side of a Triangle Given the Other Sides’ Lengths Mathematics • 11th Grade

If the measures of two sides of a triangle are 5 feet and 12 feet, what is the least possible whole-number measure for the third side?

05:07

Video Transcript

If the measures of two sides of a triangle are five feet and 12 feet, what is the least possible whole-number measure for the third side?

We’ve been given a description of a triangle for which two side lengths are known and the third side length is unknown. It’s important to recognize that the third side length could vary greatly. Let’s imagine that the two sides of length five and 12 meet at a single point; we’ll call that vertex 𝐴. The angle created at vertex 𝐴 could be an obtuse angle, a right angle, or an acute angle. Given these three possible scenarios, we should notice that as angle 𝐴 gets smaller, the length of the third side gets shorter. If angle 𝐴 was a right angle, then we could actually find the exact length of the third side. This would be achieved by using the Pythagorean theorem, which says in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.

This theorem is often represented by the equation 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the hypotenuse. If we worked out the steps of the Pythagorean theorem, we would find out that the third side length would be exactly 13. So, we have found if the measure of angle 𝐴 is 90 degrees, then the third side length of the triangle is exactly 13 feet. Let’s recall the inequality in one triangle, side comparison. When two angles have unequal measures, the side opposite the larger angle is larger than the side opposite the smaller angle. This means that if the measure of angle 𝐴 is greater than 90 degrees, the opposite side has to be greater than 13 feet. And even more importantly, if the measure of angle 𝐴 is less than 90 degrees, then the opposite side must be less than 13 feet.

To answer our original question, we are looking for the least possible whole-number measure for the third side. So we can be sure that it is going to be less than 13. But does that mean that 12 is the least possible or maybe 11 or 10? We’re going to have to do some more reasoning to figure out the true least possible whole-number measure. This is when the triangle inequality rule will be useful. According to this rule, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If we let the third side length be 𝑋, this means that five plus 12 is greater than 𝑋, 𝑋 plus five is greater than 12, and 𝑋 plus 12 is greater than five.

By simplifying each of these inequalities, we will gain more information about our third side. First, we discover that 17 is greater than 𝑋. And if we turn this inequality around, we have 𝑋 less than 17. Subtracting five from each side of our next inequality gives us 𝑋 greater than seven. Solving the third inequality gives us 𝑋 greater than negative seven. Now, that doesn’t really add any extra information because 𝑋 is a side length. By definition, it must be positive, so of course 𝑋 would be greater than negative seven. However, if we pull the first two inequalities together, we get a very important double-sided inequality.

This tells us that 𝑋 is restricted to the range of values greater than seven and less than 17. According to the original question, we are looking for the least possible whole number. Let’s make a list of whole numbers that are greater than seven and less than 17. Keep in mind that seven and 17 are not included. This includes eight, nine, 10, 11, 12, 13, 14, 15, and 16. We already knew that the third side length had to be shorter than 13 feet. But if we narrow our list all the way down to the smallest whole-number possibility, that leaves us with eight.

In conclusion, given a triangle with two side lengths of five feet and 12 feet, the least possible whole-number measure for the third side is eight feet.

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