Special Right Triangles

45-45-90 and 30-60-90 triangles

Special Right Triangles

45-45-90 Triangle

An isosceles right triangle with angles 45°, 45°, and 90°.

Side Ratios: x:x:x2x : x : x\sqrt{2}

Where:

  • xx = length of each leg
  • x2x\sqrt{2} = length of hypotenuse

Key Pattern: If legs = 1, then hypotenuse = 2\sqrt{2}

Example: If each leg is 5, hypotenuse = 525\sqrt{2}

30-60-90 Triangle

A right triangle with angles 30°, 60°, and 90°.

Side Ratios: x:x3:2xx : x\sqrt{3} : 2x

Where:

  • xx = length of side opposite 30° (shortest side)
  • x3x\sqrt{3} = length of side opposite 60°
  • 2x2x = length of hypotenuse (opposite 90°)

Key Pattern: If short leg = 1, then long leg = 3\sqrt{3}, hypotenuse = 2

Example: If short leg is 6, long leg = 636\sqrt{3}, hypotenuse = 12

Why These Are Useful

  • Appear frequently in geometry problems
  • Can solve without Pythagorean Theorem
  • Used in trigonometry
  • Found in regular polygons and circles

Remember

45-45-90: legs equal, hypotenuse = leg × 2\sqrt{2}

30-60-90: hypotenuse = 2 × short leg, long leg = short leg × 3\sqrt{3}

📚 Practice Problems

1Problem 1easy

❓ Question:

In a 45-45-90 triangle, each leg has length 8. Find the hypotenuse.

💡 Show Solution

In a 45-45-90 triangle, the hypotenuse = leg × 2\sqrt{2}

Hypotenuse=82\text{Hypotenuse} = 8\sqrt{2}

Answer: 828\sqrt{2} (or approximately 11.31)

2Problem 2medium

❓ Question:

In a 30-60-90 triangle, the hypotenuse is 20. Find both legs.

💡 Show Solution

In a 30-60-90 triangle, sides are in ratio x:x3:2xx : x\sqrt{3} : 2x

Hypotenuse = 2x2x: 2x=202x = 20 x=10x = 10

Short leg (opposite 30°): x=10x = 10

Long leg (opposite 60°): x3=103x\sqrt{3} = 10\sqrt{3}

Answer: Short leg = 1010, Long leg = 10310\sqrt{3}

3Problem 3hard

❓ Question:

A 45-45-90 triangle has a hypotenuse of 12212\sqrt{2}. Find the area of the triangle.

💡 Show Solution

Step 1: Find the leg length

In 45-45-90 triangle: hypotenuse = leg × 2\sqrt{2} leg×2=122\text{leg} \times \sqrt{2} = 12\sqrt{2} leg=12\text{leg} = 12

Step 2: Find area (both legs are equal) A=12×base×heightA = \frac{1}{2} \times \text{base} \times \text{height} A=12×12×12A = \frac{1}{2} \times 12 \times 12 A=72A = 72

Answer: Area = 7272 square units

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