SOLUTION: 30 60 90 triangle practice 1w3mynu - Studypool - Free Printable

Let’s solve each problem one by one. These are all right triangles with special angles (30° or 60°), so we can use the properties of 30-60-90 triangles.

In a 30-60-90 triangle, the sides are always in this ratio:
- Side opposite 30° → shortest leg = x
- Side opposite 60° → longer leg = x√3
- Hypotenuse (opposite 90°) = 2x

We’ll apply this to each problem.

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Problem 11:

Triangle has:
- Right angle at bottom left
- Angle at top left = 60° → so angle at bottom right = 30°
- Hypotenuse = 44
- Side adjacent to 60° (bottom side) = x
- Side opposite 60° (left side) = y

Since hypotenuse = 2x (in our ratio), then:

→ 2x = 44 → x = 22

Then, side opposite 60° = x√3 = 22√3 → that’s y

Wait — let’s double-check which side is which.

Actually, in standard labeling for 30-60-90:

If the 30° angle is at the bottom right, then:

- Side opposite 30° = short leg = y? No — wait, look again.

The triangle:

Top vertex: 60°
Bottom left: right angle (90°)
Bottom right: must be 30°

So:

- Side opposite 30° (at bottom right) = vertical side = y → this is the SHORT LEG
- Side opposite 60° (at top) = horizontal side = x → this is the LONG LEG
- Hypotenuse = 44

In 30-60-90:

Short leg (opp 30°) = x
Long leg (opp 60°) = x√3
Hypotenuse = 2x

Here, hypotenuse = 44 → 2x = 44 → x = 22

So short leg (y) = 22
Long leg (x) = 22√3

But wait — in the diagram, they labeled:

Vertical side = y
Horizontal side = x
Angle at top = 60° → so side opposite 60° is the horizontal side → that should be long leg → x = long leg = x√3? Confusing notation.

Let me redefine using variables from the diagram.

Given:

- Angle at top = 60°
- Right angle at bottom left
- So angle at bottom right = 30°
- Hypotenuse = 44 (side between top and bottom right)
- Vertical side (from top to bottom left) = y → this is adjacent to 60°, opposite to 30° → so it's the SHORT LEG
- Horizontal side (from bottom left to bottom right) = x → opposite to 60° → LONG LEG

So:

Short leg (y) = ?
Long leg (x) = ?
Hypotenuse = 44

In 30-60-90:

Short leg : Long leg : Hypotenuse = 1 : √3 : 2

So if hypotenuse = 2 parts = 44 → 1 part = 22

Thus:

Short leg (y) = 22
Long leg (x) = 22√3

✔ Final for #11:
x = 22√3
y = 22

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Problem 12:

Triangle:

Right angle at bottom right
Angle at top = 30° → so angle at bottom left = 60°
Side adjacent to 30° (vertical side) = 29√3
Hypotenuse = u
Base (horizontal) = v

Labeling:

Top vertex: 30°
Bottom right: 90°
Bottom left: 60°

So:

- Side opposite 30° = base = v → SHORT LEG
- Side opposite 60° = vertical side = 29√3 → LONG LEG
- Hypotenuse = u

In 30-60-90:

Short leg = x
Long leg = x√3
Hypotenuse = 2x

Here, long leg = 29√3 = x√3 → so x = 29

Then:

Short leg (v) = x = 29
Hypotenuse (u) = 2x = 58

✔ Final for #12:
u = 58
v = 29

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Problem 13:

Triangle:

Right angle at bottom right
Angle at bottom left = 60° → so angle at top = 30°
Vertical side = 36 → this is opposite 60° → LONG LEG
Hypotenuse = a
Base = b → opposite 30° → SHORT LEG

So:

Long leg = 36 = x√3 → x = 36 / √3 = 12√3 (rationalize: 36/√3 * √3/√3 = 36√3 / 3 = 12√3)

Wait — better to keep as:

x = 36 / √3 = 12√3? Let’s compute numerically later, but algebraically:

Set short leg = x
Then long leg = x√3 = 36 → x = 36 / √3 = 12√3? Actually:

36 / √3 = (36 √3) / 3 = 12√3 → yes.

But actually, we want exact values.

So:

Short leg (b) = x = 36 / √3 = 12√3? Wait no:

If long leg = x√3 = 36 → then x = 36 / √3

Rationalize: (36 / √3) * (√3 / √3) = (36√3)/3 = 12√3

So short leg b = 12√3

Hypotenuse a = 2x = 2*(12√3) = 24√3

But wait — is that correct?

Alternative approach: since long leg = 36, and in ratio long leg = x√3, hypotenuse = 2x

So from long leg: x√3 = 36 → x = 36/√3 = 12√3

Then hypotenuse = 2x = 24√3

Short leg = x = 12√3

But let’s check: is short leg really 12√3? That seems large.

Wait — maybe I mixed up.

In triangle:

Angle at bottom left = 60° → so side opposite to it is the vertical side = 36 → that’s correct, long leg.

Side opposite 30° (top angle) is base = b → short leg.

Yes.

So:

b = short leg = x
36 = long leg = x√3 → x = 36/√3 = 12√3

a = hypotenuse = 2x = 24√3

But 12√3 is about 20.78, and 24√3 ≈ 41.57, and 36 is given — does Pythagoras hold?

Check: (12√3)^2 + 36^2 =? (24√3)^2

Left: (144 * 3) + 1296 = 432 + 1296 = 1728
Right: (576 * 3) = 1728 → yes, correct.

So:

a = 24√3
b = 12√3

But perhaps simplify or write differently? The problem might expect simplified radical form.

Alternatively, sometimes people write without radicals in denominator, but here it’s fine.

Wait — another way: since long leg = 36, and in 30-60-90, long leg = (√3/2) * hypotenuse? Let’s see.

Standard ratios:

If hypotenuse = H, then:

Short leg = H/2
Long leg = (H√3)/2

So here, long leg = 36 = (H√3)/2 → H = 36 * 2 / √3 = 72 / √3 = 24√3 → same as before.

Short leg = H/2 = 12√3

So yes.

✔ Final for #13:
a = 24√3
b = 12√3

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Problem 14:

Triangle:

Right angle at bottom left
Angle at bottom right = 30° → so angle at top = 60°
Vertical side = 4√3 → this is opposite 30°? Let’s see.

Vertices:

Top: ?
Bottom left: 90°
Bottom right: 30° → so top angle = 60°

Sides:

Vertical side (from top to bottom left) = 4√3 → this is adjacent to 30°, opposite to 60° → so it’s the LONG LEG

Horizontal side (base) = y → opposite 30° → SHORT LEG
Hypotenuse = x

So:

Long leg = 4√3 = x√3 (if short leg = x) → so x = 4

Then:

Short leg (y) = x = 4
Hypotenuse (x) = 2x = 8

Wait — variable conflict: they used x for hypotenuse, but in ratio I used x for short leg.

Let me use different letter.

Let short leg = s
Then long leg = s√3 = 4√3 → so s = 4

Then hypotenuse = 2s = 8

So:

y = short leg = 4
x = hypotenuse = 8

✔ Final for #14:
x = 8
y = 4

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Now, problems 15–18 are too blurry to read accurately. Since the user didn’t specify which ones to solve, and only 11–14 are clear, I’ll assume we’re to solve 11–14.

Final Answers:

11) x = 22√3, y = 22
12) u = 58, v = 29
13) a = 24√3, b = 12√3
14) x = 8, y = 4

But let me write them clearly per problem.

For problem 11:
x = 22√3
y = 22

Problem 12:
u = 58
v = 29

Problem 13:
a = 24√3
b = 12√3

Problem 14:
x = 8
y = 4

I think that’s all.

Final Answer:
11) x = 22√3, y = 22
12) u = 58, v = 29
13) a = 24√3, b = 12√3
14) x = 8, y = 4