Where:

• $x$ = length of each leg
• $x\sqrt{2}$ = length of hypotenuse

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

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

30-60-90 Triangle

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

Side Ratios: $x : x\sqrt{3} : 2x$

Where:

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

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

Example: If short leg is 6, long leg = $6\sqrt{3}$, 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 × $\sqrt{2}$

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

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

Answer: $8\sqrt{2}$ (or approximately 11.31)

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

Answer: Area = $72$ square units