2.3 Special Triangles

 Special Triangles

• Definition: Special triangles are right triangles with specific angle measures that produce consistent side length ratios, making calculations easier.
• Types:
  • 45-45-90 Triangle: An isosceles right triangle with angles 45°, 45°, and 90°.
  • 30-60-90 Triangle: A right triangle with angles 30°, 60°, and 90°.
• Why They Matter: These triangles appear frequently in geometry, trigonometry, and real-world applications (e.g., engineering, architecture).

1. 45-45-90 Triangles (Isosceles Right Triangles)

Properties:

• Two equal legs (), one hypotenuse.

• Angles: 45°, 45°, 90°.

• Side Ratios: .

Diagram Description:
Draw a right triangle with two equal legs labeled and hypotenuse labeled . Mark both non-right angles as 45°.

Example 1:
Problem: A 45-45-90 triangle has legs of 7 cm. Find the hypotenuse.
Solution:
.

Example 2:
Problem: The hypotenuse of a 45-45-90 triangle is . Find the legs.
Solution:
.

2. 30-60-90 Triangles

Properties:

• Sides: Short leg (), long leg (), hypotenuse ().

• Angles: 30°, 60°, 90°.

• Side Ratios: .

Diagram Description:
Draw a right triangle with sides labeled:

• Short leg () opposite 30°.

• Long leg () opposite 60°.

• Hypotenuse ().

Example 1:
Problem: In a 30-60-90 triangle, the shorter leg is 6. Find the hypotenuse and longer leg.
Solution:

• Hypotenuse .

• Longer leg .

Example 2:
Problem: The longer leg of a 30-60-90 triangle is . Find the shorter leg.
Solution:
.
Hypotenuse .

3. Equilateral Triangles

Properties:

• All sides equal (), all angles 60°.

• Splitting an equilateral triangle along its altitude creates two 30-60-90 triangles.

Diagram Description:
Draw an equilateral triangle with side . Split it vertically into two congruent right triangles. Label:

• Base of each right triangle: .

• Height: .

Example:
Problem: An equilateral triangle has side length 10. Find its height.
Solution:
Height .

4. Summary of Key Ratios

Practice Problems

1. A 45-45-90 triangle has a hypotenuse of . Find the legs.

2. The longer leg of a 30-60-90 triangle is . Find the hypotenuse.

3. Find the area of an equilateral triangle with side length 8.