Triangle Segments Using Trig

Triangle Segments Using Trigonometry 🔺

The Story of Triangle Helpers

Imagine you have a triangle made of stretchy rubber bands. Inside this triangle, you can draw special lines that help you understand it better. These lines are like helpers that tell you secrets about the triangle!

Today, we’ll learn about four amazing helpers:

1. Medians - The balancing ropes
2. Angle Bisectors - The fair dividers
3. Altitudes - The height measurers
4. Pedal Triangle - The secret triangle hiding inside

Let’s use a pizza analogy throughout. Think of your triangle as a pizza slice!

1. Median Length Formula 📏

What is a Median?

A median is a line drawn from one corner of a triangle to the exact middle of the opposite side.

Pizza Example: Imagine you want to balance your pizza slice on your finger. The median is like finding the perfect balancing point!

The Magic Formula

If your triangle has sides a, b, and c, and you want to find the median m_a (from corner A to the middle of side a):

m_a = (1/2) × √(2b² + 2c² - a²)

Simple Example

Problem: A triangle has sides a = 6, b = 4, c = 4. Find the median to side a.

Solution:

m_a = (1/2) × √(2×4² + 2×4² - 6²)
m_a = (1/2) × √(32 + 32 - 36)
m_a = (1/2) × √28
m_a = (1/2) × 5.29
m_a ≈ 2.65

Visual Flow

graph TD
    A["Triangle ABC"] --> B["Pick a vertex"]
    B --> C["Find midpoint of opposite side"]
    C --> D["Draw line connecting them"]
    D --> E[That's your MEDIAN!]

Why Does This Work?

The formula uses the Pythagorean theorem in a clever way. It adds up information about two sides and subtracts information about the third side.

Remember: Every triangle has 3 medians, and they ALL meet at one point called the centroid (the balance point)!

2. Angle Bisector Length Formula ✂️

What is an Angle Bisector?

An angle bisector is a line that cuts an angle exactly in half - like cutting a pizza slice into two EQUAL smaller slices!

The Magic Formula

For a triangle with sides a, b, c, the length of the angle bisector t_a from angle A to side a is:

t_a = (2bc × cos(A/2)) / (b + c)

Alternate Formula (using sides only):

t_a = (2/(b+c)) × √(bc × s × (s-a))

where s = (a+b+c)/2 (half the perimeter)

Simple Example

Problem: In triangle ABC, b = 5, c = 3, and angle A = 60°. Find the angle bisector from A.

Solution:

t_a = (2 × 5 × 3 × cos(30°)) / (5 + 3)
t_a = (30 × 0.866) / 8
t_a = 25.98 / 8
t_a ≈ 3.25

Visual Flow

graph TD
    A["Start at angle A"] --> B["Measure the angle"]
    B --> C["Divide by 2"]
    C --> D["Draw line to opposite side"]
    D --> E[That's your ANGLE BISECTOR!]

Cool Fact!

The angle bisector divides the opposite side in the ratio of the adjacent sides.

If the bisector from A hits side a at point D:

BD/DC = c/b

Pizza Translation: If you cut your pizza angle in half, the crust pieces won’t be equal unless the two sides of your slice are the same length!

3. Altitude Using Sine 📐

What is an Altitude?

An altitude is a line from a corner straight DOWN to the opposite side, making a perfect 90° angle. It’s like dropping a ball from the corner - it falls straight down!

The Magic Formula

The altitude h_a from vertex A to side a:

h_a = b × sin(C) = c × sin(B)

Or in terms of area:

h_a = (2 × Area) / a

Simple Example

Problem: In a triangle, side a = 10, side b = 8, and angle C = 30°. Find the altitude to side a.

Solution:

h_a = b × sin(C)
h_a = 8 × sin(30°)
h_a = 8 × 0.5
h_a = 4

Visual Flow

graph TD
    A["Pick a vertex"] --> B["Look at opposite side"]
    B --> C["Drop straight down at 90°"]
    C --> D["Height = side × sin angle"]
    D --> E[That's your ALTITUDE!]

Why Sine?

Remember SOH from SOH-CAH-TOA?

• Sine = Opposite / Hypotenuse

In our triangle, the altitude creates a right triangle. The altitude IS the opposite side, and one of our triangle’s sides becomes the hypotenuse!

Three Altitudes, One Point!

All three altitudes of a triangle meet at one point called the orthocenter.

• In an acute triangle: orthocenter is INSIDE
• In a right triangle: orthocenter is AT the right angle corner
• In an obtuse triangle: orthocenter is OUTSIDE!

4. Properties of Pedal Triangle 🔻

What is a Pedal Triangle?

Drop altitudes from ALL three corners. Where each altitude hits the opposite side, mark a point. Connect these three points - you get a new triangle INSIDE your original one!

This hidden triangle is called the PEDAL TRIANGLE (also called the orthic triangle).

Amazing Properties

Property 1: Side Lengths

Each side of the pedal triangle equals:

Side = Original side × |cos(opposite angle)|

Example: If original side BC = 10 and angle A = 60°:

Pedal side opposite to A = 10 × cos(60°) = 10 × 0.5 = 5

Property 2: Angles

The angles of the pedal triangle are:

Angle at foot of h_a = 180° - 2A
Angle at foot of h_b = 180° - 2B
Angle at foot of h_c = 180° - 2C

Property 3: Area Relationship

Area of Pedal Triangle = Area of Original × |cos(A) × cos(B) × cos(C)| × 2

More precisely:

Area(pedal) = Area(original) × 2|cos(A)cos(B)cos(C)|

Property 4: The Reflection Trick

If you stand at the orthocenter and look at the pedal triangle, each side appears to reflect the altitude perfectly!

Visual Flow

graph TD
    A["Original Triangle ABC"] --> B["Draw altitude from A"]
    A --> C["Draw altitude from B"]
    A --> D["Draw altitude from C"]
    B --> E["Mark foot point D"]
    C --> F["Mark foot point E"]
    D --> G["Mark foot point F"]
    E --> H["Connect D, E, F"]
    F --> H
    G --> H
    H --> I["PEDAL TRIANGLE!"]

Simple Example

Problem: Triangle ABC has angles A = 50°, B = 60°, C = 70°. Find the angles of its pedal triangle.

Solution:

Angle at foot of altitude from A = 180° - 2(50°) = 80°
Angle at foot of altitude from B = 180° - 2(60°) = 60°
Angle at foot of altitude from C = 180° - 2(70°) = 40°

Check: 80° + 60° + 40° = 180° ✓

Quick Summary Table

The Big Picture 🎯

All these triangle helpers are connected!

graph TD
    A["TRIANGLE"] --> B["Medians meet at CENTROID"]
    A --> C["Angle Bisectors meet at INCENTER"]
    A --> D["Altitudes meet at ORTHOCENTER"]
    D --> E["Altitude feet form PEDAL TRIANGLE"]

Remember:

• Medians = Balance (Centroid)
• Angle Bisectors = Fairness (Incenter)
• Altitudes = Height (Orthocenter)
• Pedal Triangle = Hidden treasure inside!

You’ve Got This! 💪

These formulas might look scary, but they’re just recipes! Each one tells you exactly what ingredients (sides and angles) you need and how to mix them.

Practice Tip: Start with the altitude formula (simplest!), then try medians, then angle bisectors, and finally explore pedal triangles.

Every triangle has these helpers hiding inside, waiting to be discovered. Now YOU know how to find them! 🌟