The answer to this question depends on a number of things. First, is the triangle a right triangle, meaning at least one angle is 90 degrees? If it is a right triangle and we know the length of two sides, we can use the Pythagorean theorem to determine the third. The Pythagorean theorem tells us that:
a2 + b2 = c2
However, we also need to know whether 10 is the long side of the triangle opposite the 90 degree angle, which is called the hypotenuse, or if it is one of the shorter sides, called the legs.
Let's first assume 4 and 10 represent the value of the legs. In this case, we know a = 4 and b = 10, so we can determine c. 42 = 16 and 102 = 100. The sum of 16 + 100 = 116. The square root of 116 is approximately 10.77, which represents the hypotenuse.
On the other hand, if 10 represents the hypotenuse and 4 represents one of the legs, we know a = 4 and c = 10, with the equation represented like so:
c2 - a2 = b2
In this case, 102 - 42 can be represented as 100 - 16 = 84, and 84 = b2. The square root of 84 is approximately 9.165, which is the value of the second leg.
If the triangle is NOT a right triangle, but perhaps a scalene or isosceles triangle, for instance, we have to use the law of cosines to define the relationship between angles and sides.
Webster's defines cosine as "a trigonometric function that for an acute angle is the ratio between the leg adjacent to the angle when it is considered part of a right triangle and the hypotenuse."
The law of cosines reads as follows:
a2 = b2 + c2 - 2bc * cosA
In other words, side a2 must equal the side b2 plus c2 minus twice the product of b times c times the cosine of the angle between sides b and c.
That said, to determine the third side of a non-right triangle, you need to be given any combination of the following:
1.) Two sides AND the angle between them
2.) Two angles AND any one side
The problem here is that it is impossible to determine the length of the third side of a non-right triangle without knowing how far apart they diverge from their connecting point. Once the cosine of the angle is provided, a concrete value can be determined for the third side.
https://www.mathsisfun.com/pythagoras.html
https://www.merriam-webster.com/dictionary/cosine
https://www.youtube.com/watch?v=1rXFbSCm67U
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