Understanding Triangle Theorems

What are Triangle Theorems?

Triangle theorems are fundamental principles in geometry that describe the relationships between the sides, angles, and other properties of triangles. These theorems form the basis for solving various geometric problems and have applications in many fields, including architecture, engineering, and physics.

Key Triangle Theorems

1. Pythagorean Theorem

   For a right triangle with sides a, b, and hypotenuse c:

   \[a^2 + b^2 = c^2\]

   This theorem is used to find the length of an unknown side in a right triangle when two sides are known.

2. Law of Sines

   For any triangle with sides a, b, c and opposite angles A, B, C:

   \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

   This law is useful when we know a side and its opposite angle, plus one other side or angle.

3. Law of Cosines

   For any triangle with sides a, b, c and angle C opposite side c:

   \[c^2 = a^2 + b^2 - 2ab \cos C\]

   This law is used when we know three sides, or two sides and the included angle.

Example Calculation

Let's solve a triangle using the Law of Cosines:

Given: a = 5, b = 7, c = 8

1. Find angle C: \[C = \arccos(\frac{a^2 + b^2 - c^2}{2ab})\] \[C = \arccos(\frac{5^2 + 7^2 - 8^2}{2(5)(7)}) \approx 82.82°\]
2. Find angle A using the Law of Sines: \[A = \arcsin(\frac{a \sin C}{c}) = \arcsin(\frac{5 \sin 82.82°}{8}) \approx 38.21°\]
3. Find angle B: \[B = 180° - A - C = 180° - 38.21° - 82.82° \approx 58.97°\]

Visual Representation

This diagram illustrates the solved triangle with its sides and angles.