How can I find the length of the third side of any triangle?

Question

I will know the length of two sides of any triangle that I use, but I will not know any of the angles. I know how to find the length of the third side if I knew the angle where I am sitting, but how can I quickly find the included angle where I am sitting with basic geometric tools or something else?

Answer 1

Do you have a compass? You could determine the bearing of each object. Find the difference. And then you have the included angle and can determine the distance.

Answer 2

Knowing only the lengths of two sides of the triangle, and no angles, you cannot calculate the length of the third side; there are an infinite number of answers.

To illustrate, imagine that you have two fixed-length pieces of wood, and you drill a hole near the end of each one and put a nail through the hole. You can then rotate the pieces relative to each other, changing the angle between them, and thus changing the distance between their free ends.

Answer 3

Basic Answer:

You can't.

Here's what you can know

Given only the length of two sides of a triangle, the length of the third side is not fixed. Let a and b represent the lengths of the two known sides such that $a \geq b$. Let c represent the length of the unknown side, the length of c must fall within

$a - b < c < a + b$

Based on the example triangle you give, the third side, c, must be

\begin{align} 475 - 390 <& c < 475 + 390 \\ 85 <& c < 865 \end{align}

More Detail

With the two given lengths, we can construct a segment and a circle. It is irrelevant where we choose to position the segment. It's endpoints can lie anywhere with the caveat that it has a length exactly equal to the first known side. For your example let the segment have length 475.

With the second side we can represent all possible endpoints as the points of a circle. This makes sense since a circle is the set of all points a given distance away from the center point. So the circle would have radius 390.

Well, you want to know the third side of the triangle, but the third side -- without any other information about the triangle -- could be any segment which starts at the free endpoint of our original segment and has its other endpoint on the circle. You can see why this means that there is more than one possible segment length because not all such segments have equal length. Here are a few examples.

Closing Remarks

If you want to calculate the third side of the triangle, you need more information than simply two sides. For example, if you know the triangle is a right triangle, or if you know the measure of the included angle between the two known segments, then you can determine the length of the third side.

Answer 4

You need more information to determine that.

If the angle between those measurements reach zero (this is, the extreme points line in a line in front of you), such unknown will be 475 - 390 = 85.

The opposite could happen too, 390 measured in one direction and 475 in the other direction. In that case that unknown would be 457 + 390 = 865.

You see, there is no formula to get you that unknown in the present setting. Any value between 85 and 865 would be reasonable depending on the shape of that triangle.

So my kind suggestion is to get hold on another piece of data. As it seems you are able to measure distances, the easiest thing to do is to move yourself to a new position (say 10 yards apart), measure the distances again to your reference points, and measure also the distance you moved. If this is possible for you, then a rephrasing of your problem would help those who want to help you here.

Answer 5

$$x + 475 > 390$$ $$390 + 475 > x $$ $$390 + x > 475$$

so $x$ can be any number bigger than 85 or smaller than 865

$85<x<865$