Special Right Triangles

What are Special Right Triangles?

Special right triangles have angle measures and side ratios that follow predictable patterns.

Two types:

• 45-45-90 triangle (Isosceles right triangle)
• 30-60-90 triangle

Why special? No need for Pythagorean Theorem—use the ratios!

45-45-90 Triangle

Angles: 45°, 45°, 90°

Properties:

• Isosceles right triangle
• Two legs are equal
• Forms half of a square

Side ratio: 1 : 1 : √2

• Legs: x
• Hypotenuse: x√2

Pattern: If leg = x, then hypotenuse = x√2

Example 1: Find Hypotenuse

45-45-90 triangle with leg = 5

Hypotenuse = 5√2

Answer: 5√2

Example 2: Find Leg from Hypotenuse

45-45-90 triangle with hypotenuse = 8

Leg = hypotenuse / √2

Leg = 8 / √2
Leg = 8√2 / 2  (rationalize)
Leg = 4√2

Answer: Each leg = 4√2

Example 3: Both Legs Known

Legs are each 6

Hypotenuse = 6√2

Answer: 6√2

Example 4: Decimal Approximation

Leg = 10

Hypotenuse = 10√2 ≈ 10(1.414) ≈ 14.14

Where 45-45-90 Triangles Appear

Diagonal of a square:

• Square with side s
• Diagonal = s√2

Example: Square with side 8

• Diagonal = 8√2

30-60-90 Triangle

Angles: 30°, 60°, 90°

Properties:

• Forms half of an equilateral triangle
• Three different side lengths

Side ratio: 1 : √3 : 2

• Short leg (opposite 30°): x
• Long leg (opposite 60°): x√3
• Hypotenuse (opposite 90°): 2x

Pattern: If short leg = x, then:

• Long leg = x√3
• Hypotenuse = 2x

Example 1: Short Leg Known

30-60-90 triangle, short leg = 4

Long leg: 4√3 Hypotenuse: 2(4) = 8

Answer: Long leg = 4√3, Hypotenuse = 8

Example 2: Hypotenuse Known

30-60-90 triangle, hypotenuse = 12

Short leg = hypotenuse / 2

Short leg = 12 / 2 = 6

Long leg = short leg × √3

Long leg = 6√3

Answer: Short leg = 6, Long leg = 6√3

Example 3: Long Leg Known

30-60-90 triangle, long leg = 9√3

Short leg = long leg / √3

Short leg = 9√3 / √3 = 9

Hypotenuse = 2 × short leg

Hypotenuse = 2(9) = 18

Answer: Short leg = 9, Hypotenuse = 18

Example 4: Decimal Values

Short leg = 5

Long leg = 5√3 ≈ 5(1.732) ≈ 8.66 Hypotenuse = 10

Where 30-60-90 Triangles Appear

Half of equilateral triangle:

• Equilateral triangle with side s
• Height = (s√3)/2

Example: Equilateral triangle, side 10

• Height = (10√3)/2 = 5√3

Identifying Which Triangle

Look at the angles:

• Two 45° angles → 45-45-90
• 30° and 60° angles → 30-60-90

Or look at the sides:

• Two equal sides → 45-45-90
• Sides in ratio 1:√3:2 → 30-60-90

Comparison Table

Working Backwards

Given two sides, identify the triangle type

Example: Find Missing Angle

Sides: 5, 5√2, and unknown angle at the side 5

Recognize: Two sides are x and x√2 → 45-45-90

The angle opposite 5: 45°

Finding Area

45-45-90 triangle:

• Area = (1/2) × leg × leg = (1/2)x²

30-60-90 triangle:

• Area = (1/2) × short leg × long leg
• Area = (1/2) × x × x√3 = (x²√3)/2

Example: Area of 45-45-90

Leg = 6

Area = (1/2)(6)(6) = 18

Example: Area of 30-60-90

Short leg = 4

Long leg = 4√3

Area = (1/2)(4)(4√3) = 8√3

Rationalizing Denominators

When dividing by √2 or √3:

Example: x / √2

Multiply by √2/√2:
x / √2 × √2/√2 = x√2 / 2

Example: x / √3

Multiply by √3/√3:
x / √3 × √3/√3 = x√3 / 3

Real-World Applications

Architecture: Roof pitches

• 45° pitch creates 45-45-90 triangle

Construction: Ramp angles

• Standard angles often 30° or 45°

Engineering: Strength of diagonal braces

• Calculate exact lengths

Navigation: Direction changes

• 45° turns create 45-45-90 relationships

Practice