# How do we know the lengths of the sides of a triangle?

• Icebreaker
In summary, we can determine the lengths of the sides of a triangle without using Pythagoras' theorem by using elementary methods and trigonometry. This can be done by defining the length of one side and using the angles to calculate the other two sides using formulas involving polynomials. Another way to visualize this is by drawing squares with the lengths of the sides as their sides. This relationship can also be seen in "The Elements".
Icebreaker
Besides empirical measuring and without Pythagoras' knowledge of the theorem, how do we know the lengths of the sides of a triangle? Is this dealt with in "The Elements"?

That is, if we define the length of one side, with three angles, how do we know what's the measure of the other two sides using the most elementary methods?

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As far as I am aware, there are no more elementary methods for solving that kind of problem than by using trigonometry.

Suppose we are given a length L for the hypotenuse of this right triangle, and an angle a. Then the other two sides of the triangle are given by:

$$L_1 = L (a- \frac{a^3}{3!} + \frac{a^5}{5!}...)$$

$$L_2 = L (1+ \frac{a^2}{2!} - \frac{a^4}{4!}...)$$

This can be deduced from a knowledge of polynomials, and viewing the sine of the angle as a function. It would be interesting to derive these formulae by a purely geometric argument.

## 1. What is the Pythagorean theorem and how does it relate to the lengths of the sides of a triangle?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem can be used to find the length of any side of a right triangle if the lengths of the other two sides are known.

## 2. How do we find the length of the hypotenuse using the Pythagorean theorem?

To find the length of the hypotenuse, also known as the longest side of a right triangle, square the lengths of the other two sides and add them together. Then, take the square root of that sum to find the length of the hypotenuse.

## 3. Can we use the Pythagorean theorem to find the length of any side of a triangle?

No, the Pythagorean theorem can only be used to find the length of a side in a right triangle. It does not apply to triangles that do not have a right angle.

## 4. Are there any other methods to find the lengths of the sides of a triangle?

Yes, there are other methods such as using trigonometric functions (sine, cosine, and tangent) and the Law of Cosines. These methods can be used to find the length of any side of a triangle, not just right triangles.

## 5. What is the importance of knowing the lengths of the sides of a triangle?

Knowing the lengths of the sides of a triangle is important in many fields, including engineering, architecture, and navigation. It allows us to calculate distances, angles, and other important measurements that are necessary for building structures and solving real-world problems.

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