Step 1: Recall 45-45-90 triangle properties: This is an isosceles right triangle If each leg = x, then hypotenuse = x√2

Step 2: Identify the given information: One leg = 6

Step 3: Find the other leg: In a 45-45-90 triangle, both legs are equal Other leg = 6

Step 4: Find the hypotenuse: Hypotenuse = leg × √2 Hypotenuse = 6√2

Step 5: Verify using Pythagorean Theorem: 6² + 6² = (6√2)² 36 + 36 = 36 × 2 72 = 72 ✓

Answer: Other leg = 6, Hypotenuse = 6√2

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

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

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

Step 1: Recall 30-60-90 triangle ratios: If shorter leg (opposite 30°) = x Then longer leg (opposite 60°) = x√3 And hypotenuse = 2x

Step 2: Identify given information: Shorter leg = 5, so x = 5

Step 3: Find the longer leg: Longer leg = x√3 = 5√3

Step 4: Find the hypotenuse: Hypotenuse = 2x = 2(5) = 10

Step 5: Verify using Pythagorean Theorem: 5² + (5√3)² = 10² 25 + 25(3) = 100 25 + 75 = 100 100 = 100 ✓

Answer: Longer leg = 5√3, Hypotenuse = 10

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

Hypotenuse = $2x$: $2x = 20$ $x = 10$

Short leg (opposite 30°): $x = 10$

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

Answer: Short leg = $10$, Long leg = $10\sqrt{3}$

Step 1: Recall the ratio: Shorter leg : Longer leg : Hypotenuse = x : x√3 : 2x

Step 2: Use the hypotenuse to find x: Hypotenuse = 2x 20 = 2x x = 10

Step 3: Find the shorter leg: Shorter leg = x = 10

Step 4: Find the longer leg: Longer leg = x√3 = 10√3

Step 5: Verify: 10² + (10√3)² = 20² 100 + 100(3) = 400 100 + 300 = 400 400 = 400 ✓

Answer: Shorter leg = 10, Longer leg = 10√3

Step 1: Recall the relationship: In a 45-45-90 triangle: If leg = x, then hypotenuse = x√2

Step 2: Set up the equation: x√2 = 8√2

Step 3: Solve for x: x = 8√2 / √2 x = 8

Step 4: Both legs equal x: Each leg = 8

Step 5: Verify: 8² + 8² = (8√2)² 64 + 64 = 64 × 2 128 = 128 ✓

Answer: Each leg = 8

Step 1: Find the leg length

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

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

Step 1: Recognize the special triangle: A square's diagonal divides it into two 45-45-90 triangles

Step 2: Set up the relationship: In a 45-45-90 triangle: If leg = s (side of square), then hypotenuse = s√2 The diagonal is the hypotenuse

Step 3: Use the given diagonal: s√2 = 10

Step 4: Solve for s: s = 10/√2 s = 10/√2 × √2/√2 (rationalize) s = 10√2/2 s = 5√2

Step 5: Find the perimeter: Perimeter = 4s = 4(5√2) = 20√2

Step 6: Verify the diagonal: Using Pythagorean theorem: s² + s² = diagonal² (5√2)² + (5√2)² = 10² 25(2) + 25(2) = 100 50 + 50 = 100 ✓

Answer: Side length = 5√2, Perimeter = 20√2