Euclidean geometry, the basic geometry taught in school, requires certain relationships between the lengths of the sides of a triangle. One cannot simply take three random line segments and form a triangle. The line segments have to satisfy the triangle inequality theorems. Other theorems that define relationships between the sides of a triangle are the Pythagorean theorem and the law of cosines.
Triangle Inequality Theorem One
According to the first triangle inequality theorem, the lengths of any two sides of a triangle must add up to more than the length of the third side. This means that you cannot draw a triangle that has side lengths 2, 7 and 12, for instance, since 2 + 7 is less than 12. To get an intuitive feel for this, imagine first drawing a line segment 12 cm long. Now think of two other line segments 2 cm and 7 cm long attached to the two ends of the 12 cm segment. It is clear that it would not be possible to make the two end segments meet. They would have to add up at least to 12 cm.
Triangle Inequality Theorem Two
The longest side in a triangle is across from the largest angle. This is another triangle inequality theorem and it makes intuitive sense. You can draw various conclusions from it. For example, in an obtuse triangle, the longest side has to be the one across from the obtuse angle. The converse of this is true as well. The largest angle in a triangle is the one that is across from the longest side.
The Pythagorean theorem states that, in a right triangle, the square of the length of the hypotenuse (the side across from the right angle) is equal to the sum of the squares of the other two sides. So if the length of the hypotenuse is c and the lengths of the other two sides are a and b, then c^2 = a^2 + b^2. This is an ancient theorem that has been known for thousands of years and has been used by builders and mathematicians through the ages.
Law of Cosines
The law of cosines is a generalized version of the Pythagorean theorem that applies to all triangles, not just the ones with right angles. According to this law, if a triangle had sides of length a, b and c, and the angle across from the side of length c is C, then c^2 = a^2 + b^2 - 2abcosC. You can see that when C is 90 degrees, cosC = 0 and the law of cosines is reduced to the Pythagorean theorem.
- Thinkstock/Comstock/Getty Images