Geometry Chapter on Triangles

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.