30-60-90 Triangle Practice Worksheet with Diagrams and Calculations

I'll solve each 30°-60°-90° triangle problem using the special right triangle ratios.

Key relationships for 30°-60°-90° triangles:
- Short leg (opposite 30°) = n
- Long leg (opposite 60°) = n√3
- Hypotenuse = 2n

Let me work through each problem:

Problem 1:
- Given: short leg = 1/2, angles are 30° and 60°
- x is opposite 30°, so x = short leg = 1/2
- y is hypotenuse = 2 × short leg = 2 × (1/2) = 1

Problem 2:
- Given: long leg = 16, angle at bottom is 30°
- Long leg = n√3 = 16
- So n = 16/√3 = 16√3/3
- x (hypotenuse) = 2n = 2(16√3/3) = 32√3/3
- y (short leg) = n = 16√3/3

Wait, let me reconsider. Looking at the diagram more carefully:
- The right angle is at the bottom right
- The 30° angle is at the bottom left
- So y is the short leg (adjacent to 30°), x is hypotenuse
- Long leg = 16 = n√3
- n = 16/√3 = 16√3/3
- y = n = 16√3/3
- x = 2n = 32√3/3

Actually, let me be more careful with which side is which based on the angles shown.

Problem 2 (revised):
- Right angle at bottom right corner
- 30° angle at bottom left
- Side labeled 16 is vertical (long leg, opposite 60°)
- y is horizontal (short leg, opposite 30°)
- x is hypotenuse
- 16 = n√3, so n = 16/√3 = 16√3/3
- y = n = 16√3/3 ≈ 9.24
- x = 2n = 32√3/3 ≈ 18.48

Hmm, these aren't nice numbers. Let me check if I'm reading the diagram correctly. Actually, in many of these problems, the answers should be simpler. Let me reconsider whether 16 might be a different side.

Looking again at problem 2: if the vertical side is 16 and it's opposite the 30° angle, then it would be the short leg. But that doesn't match the angle markings. Let me assume the standard interpretation where the side opposite 60° is the long leg.

Actually, I think I need to look at this more systematically. Let me recalculate all problems: