Geometry Chapter on Triangles

Study Notes

Pythagorean Theorem and Triples

The Pythagorean Theorem states for right triangles, (a^2 + b^2 = c^2), where (c) is the hypotenuse.
A Pythagorean triple consists of non-zero whole numbers (a), (b), and (c) that satisfy (a^2 + b^2 = c^2).
The set (5, 12, 13) is an example of a Pythagorean triple.

Triangle Type Theorems

Theorem 8-3: If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, the triangle is obtuse.
Theorem 8-4: If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, the triangle is acute.

Specialized Triangle Theorems

Theorem 8-5: In a 45-45-90 triangle, legs are congruent, and the hypotenuse is (\sqrt{2}) times a leg's length.
Theorem 8-6: In a 30-60-90 triangle, the hypotenuse is twice the shorter leg, and the longer leg is (\sqrt{3}) times the shorter leg.

Triangle Properties

Cutting an equilateral triangle with a bisector creates two scalene 30-60-90 triangles.

Trigonometric Functions

Soh Cah Toa defines trigonometric ratios:
- Sine: (\sin = \frac{\text{Opposite}}{\text{Hypotenuse}})
- Cosine: (\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}})
- Tangent: (\tan = \frac{\text{Opposite}}{\text{Adjacent}})

Angle Calculations

To find an angle (X) in a right triangle with a known adjacent side of 7 and hypotenuse of 10:
- Use: (\tan^{-1} \left(\frac{7}{10}\right)) to get approximately 35 degrees.

Similar Right Triangles

Similar right triangles maintain equivalent ratios for their corresponding sides through trigonometric ratios.

Angles of Elevation and Depression

The angle of elevation is the angle at which an observer looks upwards.
The angle of depression is viewed when looking down at an object; these angles are equal due to alternate interior angles.

Triangle Solution Requirements

To solve a right triangle, at least two pieces of information are needed, such as two side lengths or an angle with a side length.

Description

Explore the properties and theorems of triangles with this quiz. Test your knowledge on the Pythagorean theorem, triangle type theorems, and specialized triangle characteristics. Perfect for understanding triangle fundamentals in geometry.