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Special Right Triangles

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

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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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📌 Related Topics in Triangles

Triangle Angle Sum TheoremCongruent TrianglesPythagorean TheoremSimilar TrianglesTriangle InequalitiesSpecial Segments in Triangles

❓ Frequently Asked Questions

What is Special Right Triangles?▾
45-45-90 and 30-60-90 triangles
How can I study Special Right Triangles effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 3 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Special Right Triangles study guide free?▾
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What course covers Special Right Triangles?▾
Special Right Triangles is part of the Geometry course on Study Mondo, specifically in the Triangles section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Special Right Triangles?▾
Yes, this page includes 3 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.