Special Right Triangles Calculator

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Special Right Triangles

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

How To Solve Special Right Triangles

Step 1: Identify Triangle Type

Determine if the triangle is 45-45-90 (isosceles) or 30-60-90 (30-60-90).

Why: Each type has a distinct ratio pattern; using the wrong one invalidates calculations.

Step 2: Identify Which Measurement Is Known

For 45-45-90: leg or hypotenuse? For 30-60-90: short leg, long leg, or hypotenuse?

Why: Each known element leads to a different calculation path using the fixed ratios.

Step 3: Apply the Correct Ratio Pattern

45-45-90 ratio is 1:1:√2. For 30-60-90 ratio is 1:√3:2 (short leg:long leg:hypotenuse).

Why: These fixed ratios are derived from trigonometry and are always exact for these triangles.

Step 4: Calculate Unknown Sides

Multiply or divide the known value by the appropriate ratio components.

Why: The ratios define the proportional relationships that scale to any size triangle.

Step 5: Verify All Ratios Match

Check that all three sides are consistent with the triangle type's ratio.

Why: This confirms calculations are correct and the triangle is actually the claimed type.

Detailed Example

Scenario: A 45-45-90 triangle has a leg of 5 cm. Find the other leg and hypotenuse.
Step 1 - Identify Type: 45-45-90 triangle (isosceles right triangle).
Step 2 - Identify Known: One leg = 5 cm (known).
Step 3 - Apply Ratio: 45-45-90 ratio is 1:1:√2 (leg:leg:hypotenuse).
Step 4 - Calculate: Other leg = 1 × 5 = 5 cm. Hypotenuse = 5√2 ≈ 7.07 cm.
Step 5 - Verify: Ratio check: 5:5:7.07 simplifies to 1:1:√2 ✓ Using Pythagorean theorem: 5² + 5² = 25 + 25 = 50 = (5√2)² ✓
Verification: Both legs equal (isosceles confirmed), hypotenuse = leg×√2 (ratio confirmed).
Result: The triangle has legs 5 cm, 5 cm, and hypotenuse 5√2 ≈ 7.07 cm.
Interpretation: 45-45-90 triangles are isosceles; the hypotenuse is always √2 times a leg length.

Ratios

45-45-90: Sides are in ratio 1 : 1 : √2

30-60-90: Sides are in ratio 1 : √3 : 2

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