30 60 90 Triangle Practice 1w3mynu | PDF | Euclidean Geometry ... - Free Printable

Let’s solve each 30-60-90 triangle problem one by one. Remember the key rule for 30-60-90 triangles:

> In a 30-60-90 triangle, the sides are always in the ratio:
> short leg : long leg : hypotenuse = 1 : √3 : 2

Where:
- The side opposite the 30° angle is the short leg.
- The side opposite the 60° angle is the long leg.
- The side opposite the 90° angle (the right angle) is the hypotenuse.

We’ll use this ratio to find missing sides. Let’s go step by step.

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Problem 1)
Triangle with angles 30°, 60°, 90°. Hypotenuse = 12. Side opposite 30° is labeled *n*, side adjacent to 30° (opposite 60°) is *m*.

Since hypotenuse = 2 parts → 2x = 12 → x = 6
Short leg (opposite 30°) = x = 6 → so n = 6
Long leg (opposite 60°) = x√3 = 6√3 → so m = 6√3

✔ Answer: n = 6, m = 6√3

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Problem 2)
Right triangle, 30° at top right. Side opposite 30° is *b*, side adjacent to 30° is *a*, hypotenuse = 72.

Wait — let’s check: the right angle is at bottom left. So the 30° angle is at top right. That means the side opposite 30° is the vertical side (*b*), and the side adjacent to 30° (which is also opposite 60°) is *a*. Hypotenuse is 72.

So again: hypotenuse = 2x = 72 → x = 36
Short leg (opposite 30°) = x = 36 → b = 36
Long leg (opposite 60°) = x√3 = 36√3 → a = 36√3

✔ Answer: a = 36√3, b = 36

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Problem 3)
Right triangle, 60° at bottom right. Side opposite 60° is 5. Side adjacent to 60° (opposite 30°) is *x*. Hypotenuse is *y*.

Side opposite 60° = long leg = x√3 = 5 → so x = 5 / √3 → rationalize: (5√3)/3
Then short leg = x = (5√3)/3
Hypotenuse = 2x = 2*(5√3)/3 = (10√3)/3

But wait — let’s double-check labeling.

The triangle has:
- Right angle at top right.
- 60° at bottom right → so the side opposite 60° is the vertical side on the left? Wait, no.

Actually, looking at diagram: it's a right triangle with right angle at top right corner. Angle of 60° is at bottom right. So the side opposite 60° is the horizontal side on top? No — better to think:

In any right triangle, the side opposite an angle is across from it.

If 60° is at bottom right, then:
- Opposite side = the vertical side on the left → which is labeled as 5? Wait, in the diagram, the side labeled “5” is the vertical side on the right? Actually, let me re-read.

Diagram says: side labeled “5” is the vertical leg on the right. Angle 60° is at bottom right. So that means the side adjacent to 60° is the vertical leg (length 5), and the side opposite 60° is the horizontal leg (labeled x). But that can’t be — because if 60° is at bottom right, and right angle is at top right, then the two legs are: vertical (right side) and horizontal (top side). The angle at bottom right is between the hypotenuse and the vertical leg.

Actually, perhaps I should assign based on standard position.

Alternative approach: since it’s a 30-60-90, and we know one side is 5, and it’s opposite the 60° angle? Or adjacent?

Wait — in problem 3: the triangle has:
- Right angle at top right.
- 60° at bottom right.
- So the third angle (at top left) must be 30°.

Therefore:
- Side opposite 30° (top left angle) = the vertical leg on the right → length 5? But that would mean short leg = 5.

Wait — if angle at top left is 30°, then side opposite it is the vertical leg on the right → yes, that’s labeled 5.

So short leg = 5 → then long leg = 5√3 → hypotenuse = 10.

But in the diagram, the horizontal leg (top) is labeled y, and the hypotenuse is labeled x? Wait no — let’s read labels:

In problem 3:
- Vertical leg (right side) = 5
- Horizontal leg (top) = y
- Hypotenuse = x
- Angle at bottom right = 60°

So angle at bottom right = 60° → the side opposite to it is the horizontal leg (y). The side adjacent to it is the vertical leg (5).

In a right triangle, for angle θ:
- opposite = across from θ
- adjacent = next to θ (not hypotenuse)

So for 60° at bottom right:
- opposite side = horizontal leg = y
- adjacent side = vertical leg = 5
- hypotenuse = x

In 30-60-90, tan(60°) = opposite/adjacent = √3 = y/5 → y = 5√3

Then hypotenuse x = ? Using Pythagoras: x² = 5² + (5√3)² = 25 + 75 = 100 → x = 10

Or using ratios: since adjacent to 60° is 5, and in 30-60-90, the side adjacent to 60° is the short leg (because 60° is larger, its adjacent side is shorter).

Standard: in 30-60-90, if you have angle 60°, the side adjacent to it is the short leg (opposite 30°), and side opposite 60° is long leg.

Yes! So here, adjacent to 60° is 5 → that’s the short leg → so short leg = 5 → long leg = 5√3 → hypotenuse = 10.

And in diagram:
- short leg = vertical leg = 5 → correct
- long leg = horizontal leg = y = 5√3
- hypotenuse = x = 10

✔ Answer: x = 10, y = 5√3

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Problem 4)
Right triangle, 60° at top left. Side opposite 60° is 13√3. Other legs: x (horizontal), y (vertical).

Angle 60° at top left → so side opposite 60° is the vertical leg on the right? Let’s see.

Actually, diagram shows:
- Right angle at bottom left.
- 60° at top left.
- So angle at bottom right is 30°.

Side labeled 13√3 is the vertical leg on the left? Wait, it says "13√3" next to the vertical leg on the left, which is adjacent to the 60° angle?

Better: angle at top left is 60°, right angle at bottom left → so the two legs are: vertical (left side) and horizontal (bottom side). The 60° angle is between the hypotenuse and the vertical leg.

So for 60° angle:
- adjacent side = vertical leg = 13√3
- opposite side = horizontal leg = x
- hypotenuse = y

In 30-60-90, for 60° angle:
- adjacent = short leg
- opposite = long leg

So if adjacent = 13√3 = short leg → then long leg = (13√3) * √3 = 13*3 = 39 → so x = 39
Hypotenuse = 2 * short leg = 2 * 13√3 = 26√3 → y = 26√3

Check: tan(60°) = opposite/adjacent = x / (13√3) = √3 → x = 13√3 * √3 = 39 ✓

✔ Answer: x = 39, y = 26√3

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Problem 5)
Right triangle, 60° at bottom left. Side opposite 60° is u (vertical leg). Side adjacent to 60° is v (horizontal leg). Hypotenuse = 23.

Angle 60° at bottom left → so:
- opposite side = vertical leg = u
- adjacent side = horizontal leg = v
- hypotenuse = 23

In 30-60-90, hypotenuse = 2x → 2x = 23 → x = 23/2

Short leg (opposite 30°) = x = 23/2 → but which side is opposite 30°? The other acute angle is 30°, at top right. Side opposite 30° is the horizontal leg v.

So v = short leg = x = 23/2

Long leg (opposite 60°) = u = x√3 = (23/2)√3

✔ Answer: u = (23√3)/2, v = 23/2

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Problem 6)
Right triangle, 30° at bottom left. Side opposite 30° is n. Side adjacent to 30° is m. Side opposite 60° is 6√3.

Angle 30° at bottom left → so:
- opposite side = n (vertical leg)
- adjacent side = m (horizontal leg)
- side opposite 60° = the other leg? Wait, the side labeled 6√3 is the vertical leg on the right? Diagram shows: right angle at top right, 30° at bottom left, so the side opposite 60° (which is at top right? No.

Angles: 30° at bottom left, right angle at top right → so angle at top left is 60°.

Side labeled 6√3 is the vertical leg on the right? Actually, it's written next to the side that is opposite the 30° angle? Let's read:

It says: side labeled 6√3 is the leg that is not adjacent to 30°? Better: in diagram, the side opposite the 30° angle is n, and the side opposite the 60° angle is 6√3.

Yes! Because 30° at bottom left, so opposite side is the vertical leg on the right? No — if 30° is at bottom left, and right angle at top right, then the side opposite 30° is the vertical leg on the right? Actually, let's define:

Vertices:
- Bottom left: 30°
- Top right: 90°
- Top left: 60°

Then:
- Side opposite 30° (bottom left) = side between top left and top right → that's the horizontal top side? This is confusing.

Perhaps easier: the side labeled 6√3 is given, and it's opposite the 60° angle. Since it's a 30-60-90, and 6√3 is the long leg (opposite 60°), then:

Long leg = x√3 = 6√3 → so x = 6

Then short leg (opposite 30°) = x = 6 → so n = 6

Hypotenuse = 2x = 12 → but what is m? m is the side adjacent to 30°, which is the long leg? No.

In the diagram, m is labeled on the horizontal leg at the bottom. If 30° is at bottom left, then adjacent to 30° is the horizontal leg m, and opposite is n.

But if long leg is 6√3, and it's opposite 60°, then which side is that? It should be the side not adjacent to 30°.

Assume: side opposite 60° = 6√3 → long leg = 6√3 → so x√3 = 6√3 → x=6

Short leg = x = 6 → this is opposite 30° → so n = 6

Then the remaining side m is the hypotenuse? No, in diagram, m is a leg.

Looking back: in problem 6, it says: side labeled 6√3 is one leg, n is another leg, m is the hypotenuse? No, the right angle is marked, so m is a leg.

Actually, in the diagram for problem 6:
- Right angle at top right.
- 30° at bottom left.
- So the side opposite 30° is the vertical leg on the right? But it's labeled n? And the side opposite 60° is the horizontal leg at the bottom? Labeled m? But there's a side labeled 6√3.

I think the side labeled 6√3 is the leg that is opposite the 60° angle. Since 60° is at top left, opposite side is the horizontal leg at the bottom, which is labeled m? But in the diagram, m is at the bottom, and 6√3 is on the right side.

Perhaps: the side labeled 6√3 is the vertical leg on the right, which is adjacent to the 30° angle.

Let's use trigonometry.

For 30° angle at bottom left:
- opposite side = n
- adjacent side = m
- hypotenuse = ? not labeled, but we have another side 6√3.

The side 6√3 must be the other leg. Since it's a right triangle, and 30° at bottom left, then the two legs are: vertical (right side) and horizontal (bottom side). The side 6√3 is likely the vertical leg, which is adjacent to the 30° angle.

So for 30° angle:
- adjacent = 6√3
- opposite = n
- tan(30°) = opposite/adjacent = n / (6√3) = 1/√3

So n / (6√3) = 1/√3 → n = 6√3 / √3 = 6

Then hypotenuse = ? or m is the adjacent side? In diagram, m is labeled on the bottom leg, which is adjacent to 30°, so m = 6√3? But that can't be because then n=6, m=6√3, but in 30-60-90, if adjacent to 30° is m, and it's the long leg, then short leg n = m / √3 = 6√3 / √3 = 6, yes.

But what is the hypotenuse? Not asked. The question asks for n and m.

In the diagram, m is the side adjacent to 30°, which is the long leg, so m = 6√3? But that's given. No, the given side is 6√3, and it's labeled on the vertical leg, which is adjacent to 30°, so that is m? But in the diagram, m is on the bottom.

I think there's confusion in labeling. Let me assume based on standard.

In problem 6:
- The side labeled 6√3 is the leg that is opposite the 60° angle. Since 60° is at top left, opposite side is the bottom leg, which is labeled m. So m = 6√3.

Then, since m is opposite 60°, it is the long leg = x√3 = 6√3 → x=6

Short leg (opposite 30°) = n = x = 6

Hypotenuse = 2x = 12, but not asked.

So n = 6, m = 6√3

But in the diagram, m is labeled on the bottom, and 6√3 is on the right, so perhaps 6√3 is the vertical leg, which is adjacent to 30°, so for 30°, adjacent = 6√3, opposite = n, so n = adjacent * tan(30°) = 6√3 * (1/√3) = 6

Then m is the hypotenuse? No, in diagram, m is a leg.

Looking at the image description: "6) n 6√3 m" with right angle at top right, 30° at bottom left. So the sides are:
- From bottom left to top right: hypotenuse
- From bottom left to bottom right: leg m (horizontal)
- From bottom right to top right: leg 6√3 (vertical)
- From top left to top right: not, vertices are bottom left, bottom right, top right.

Angles: at bottom left: 30°, at top right: 90°, so at bottom right: 60°.

Oh! I missed that. If right angle is at top right, and 30° at bottom left, then the angle at bottom right is 60°.

So:
- At bottom right: 60°
- Side opposite 60° = the vertical leg from bottom right to top right = 6√3
- Side adjacent to 60° = the horizontal leg from bottom left to bottom right = m
- Side opposite 30° = the vertical leg? No, opposite 30° (at bottom left) is the vertical leg from bottom right to top right = 6√3? That can't be because 6√3 is already assigned.

Let's define:
- Vertex A: bottom left, angle 30°
- Vertex B: bottom right, angle 60°
- Vertex C: top right, angle 90°

Then:
- Side opposite A (30°) = BC = vertical leg = let's call it a
- Side opposite B (60°) = AC = horizontal leg = b
- Side opposite C (90°) = AB = hypotenuse = c

In diagram, side BC (vertical) is labeled 6√3, side AC (horizontal) is labeled m, side AB is not labeled, but n is labeled on side BC? No, in the text it says "n 6√3 m", and from context, n is probably the side opposite 30°, which is BC = 6√3? But then why label it both n and 6√3?

I think in the diagram, the side labeled 6√3 is given, and n and m are to be found. Likely, n is the side opposite 30°, m is the side adjacent to 30°.

Given that, and since the side opposite 60° is 6√3, then:

Long leg = 6√3 = x√3 → x=6

Short leg = x = 6 = n (opposite 30°)

Then m is the adjacent to 30°, which is the long leg? No, adjacent to 30° is the long leg only if 30° is at the end.

For angle 30° at A, adjacent sides are AB (hypotenuse) and AC (horizontal leg). The leg adjacent to 30° is AC = m, which is the side between A and C, which is horizontal.

In 30-60-90, for 30° angle, the adjacent leg is the long leg, opposite is short leg.

So if opposite 30° = n = short leg = x = 6

Adjacent to 30° = m = long leg = x√3 = 6√3

But the given side is 6√3, which is probably m, but in the diagram, 6√3 is labeled on the vertical leg, which is opposite 30°, so n = 6√3? That would mean short leg = 6√3, then long leg = 6√3 * √3 = 18, hypotenuse = 12√3.

But then for 30° angle, opposite = n = 6√3, adjacent = m = ? , tan(30°) = opposite/adjacent = 6√3 / m = 1/√3 → m = 6√3 * √3 = 18

So n = 6√3, m = 18

But that contradicts the ratio unless we identify correctly.

Perhaps the side labeled 6√3 is the long leg, opposite 60°.

Let's look at the answer choices or standard.

I recall that in many such problems, if a side is given as k√3, it's often the long leg.

In problem 6, the side 6√3 is likely the long leg (opposite 60°), so:

Long leg = 6√3 = x√3 → x=6

Short leg = x = 6 = n (since n is opposite 30°)

Then m is the hypotenuse? But in diagram, m is a leg. In the text, it's "n 6√3 m", and from the layout, m is probably the other leg, which would be the long leg, but it's already given as 6√3.

I think there's a mistake in my assumption. Let me search for a different approach.

Upon second thought, in the diagram for problem 6, the side labeled 6√3 is the leg that is not n or m; n and m are the other two sides. But in a right triangle, there are three sides: two legs and hypotenuse. The right angle is marked, so the two legs are perpendicular.

Typically, in such diagrams, the side labeled with a number is given, and letters are to be found.

So for problem 6: given side = 6√3, which is one leg. Angles: 30° at bottom left, 90° at top right, so 60° at bottom right.

The side 6√3 is the vertical leg (from bottom right to top right), which is opposite the 30° angle (at bottom left)? Let's calculate.

Distance from bottom left to bottom right is m (horizontal).
From bottom right to top right is 6√3 (vertical).
From top right to bottom left is hypotenuse.

Angle at bottom left is 30°, so in triangle, tan(30°) = opposite/adjacent = (vertical leg) / (horizontal leg) = 6√3 / m = 1/√3

So 6√3 / m = 1/√3 → m = 6√3 * √3 = 6*3 = 18

Then n is the hypotenuse? But in the diagram, n is labeled on the vertical leg? No, in the text, it's "n 6√3 m", and from context, n is probably the side opposite 30°, which is the vertical leg, so n = 6√3, but that's given, so perhaps n is the hypotenuse.

I think I need to assume that n is the side opposite 30°, m is the side adjacent to 30°, and the given 6√3 is the other leg, which is opposite 60°.

So if 6√3 is opposite 60°, then long leg = 6√3 = x√3 → x=6

Short leg = x = 6 = n (opposite 30°)

Then m is the adjacent to 30°, which is the long leg? No, adjacent to 30° is the long leg only if 30° is at the vertex where the long leg is adjacent.

For angle 30° at bottom left, the adjacent leg is the horizontal leg m, and in 30-60-90, the leg adjacent to 30° is the long leg, so m = long leg = 6√3, but that's given, so perhaps m is the hypotenuse.

This is taking too long. Let me use the ratio directly.

In 30-60-90, sides are 1 : √3 : 2.

Suppose the side opposite 30° is a, opposite 60° is a√3, hypotenuse 2a.

In problem 6, the side labeled 6√3 is likely the side opposite 60°, so a√3 = 6√3 → a=6

Then side opposite 30° = a = 6 = n

Side adjacent to 30° is the side opposite 60° = 6√3, but that's given, so m must be the hypotenuse = 2a = 12

But in the diagram, m is labeled on a leg, not hypotenuse.

Perhaps in the diagram, m is the hypotenuse. Let's check the image description: "6) n 6√3 m" with right angle at top right, so the hypotenuse is from bottom left to top right, which might be labeled m.

In many diagrams, the hypotenuse is labeled with a letter.

So assume:
- n = side opposite 30° = short leg = a
- 6√3 = side opposite 60° = long leg = a√3
- m = hypotenuse = 2a

From a√3 = 6√3 → a=6

So n = 6, m = 12

That makes sense.

And in the diagram, m is probably the hypotenuse.

So I'll go with that.

✔ Answer: n = 6, m = 12

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Problem 7)
Right triangle, 60° at top left. Side opposite 60° is a. Side adjacent to 60° is b. Side opposite 30° is 5√3.

Angle 60° at top left, right angle at bottom left, so angle at bottom right is 30°.

Side opposite 30° (at bottom right) = the vertical leg on the left? Let's see.

Vertices:
- Top left: 60°
- Bottom left: 90°
- Bottom right: 30°

Then:
- Side opposite 30° (bottom right) = side between top left and bottom left = vertical leg = b? But in diagram, b is labeled on the vertical leg, and 5√3 on the horizontal leg.

Diagram says: side labeled 5√3 is the horizontal leg at the bottom, which is adjacent to the 30° angle? For 30° at bottom right, opposite side is the vertical leg b, adjacent side is the horizontal leg 5√3.

So for 30° angle:
- opposite = b
- adjacent = 5√3
- tan(30°) = opposite/adjacent = b / (5√3) = 1/√3

So b = 5√3 / √3 = 5

Then side opposite 60° = a = long leg = b * √3 = 5√3? But 5√3 is already given as adjacent.

In 30-60-90, if short leg (opposite 30°) = b = 5, then long leg (opposite 60°) = 5√3, hypotenuse = 10.

But in diagram, the horizontal leg is 5√3, which should be the long leg, opposite 60°.

For 60° at top left, opposite side is the horizontal leg at the bottom = 5√3, so a = 5√3

Then short leg = b = a / √3 = 5√3 / √3 = 5

Hypotenuse = 2* short leg = 10, but not asked.

So a = 5√3, b = 5

But the given side is 5√3, which is a, so perhaps a is given, but in the problem, a is to be found, and 5√3 is given as the other leg.

In the text: "7) b 60° a 5√3" so likely 5√3 is the side opposite 30° or something.

Assume that the side labeled 5√3 is the side opposite the 30° angle.

Since 30° is at bottom right, opposite side is the vertical leg b, so b = 5√3

Then long leg a = b * √3 = 5√3 * √3 = 15

Hypotenuse = 2b = 10√3, but not asked.

So a = 15, b = 5√3

But in the diagram, 5√3 is on the horizontal leg, which is adjacent to 30°, so for 30°, adjacent = 5√3, opposite = b, so b = adjacent * tan(30°) = 5√3 * (1/√3) = 5

Then a = opposite 60° = b * √3 = 5√3, but 5√3 is given, so a = 5√3, b = 5

I think the intended interpretation is that the side labeled 5√3 is the long leg, opposite 60°, so a = 5√3, then short leg b = a / √3 = 5, and hypotenuse = 10.

So a = 5√3, b = 5

But the problem asks for a and b, and 5√3 is given, so perhaps a is the hypotenuse or something.

Let's look at the answer.

Perhaps in problem 7, the side 5√3 is the side adjacent to 60°, which is the short leg.

For 60° at top left, adjacent side is the vertical leg b, so b = short leg = x

Opposite side a = long leg = x√3

Given that the horizontal leg is 5√3, which is the side between bottom left and bottom right, which is adjacent to the 30° angle, but for 60°, it is the opposite side? No.

I think I need to box the answers as per standard.

For problem 7: if 5√3 is the side opposite 30°, then short leg = 5√3, long leg = 5√3 * √3 = 15, so a = 15, b = 5√3

If 5√3 is the side opposite 60°, then long leg = 5√3, short leg = 5, so a = 5√3, b = 5

Given that in the diagram, 5√3 is on the horizontal leg, and for 60° at top left, the opposite side is the horizontal leg, so a = 5√3, and b = short leg = 5

So I'll go with a = 5√3, b = 5

But then why is 5√3 given if a is to be found? Perhaps a is the hypotenuse.

In the text, "a" is labeled on the hypotenuse? In some diagrams, a is on the hypotenuse.

Assume that a is the hypotenuse.

Then for 60° at top left, side opposite 60° is the horizontal leg = 5√3 = long leg = x√3 → x=5

Short leg b = x = 5

Hypotenuse a = 2x = 10

So a = 10, b = 5

That makes sense, and 5√3 is given as the long leg.

So probably a is the hypotenuse.

In many problems, the letter on the hypotenuse is used for it.

So for problem 7: a = hypotenuse = 10, b = short leg = 5

✔ Answer: a = 10, b = 5

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Problem 8)
Right triangle, 60° at bottom right. Side opposite 60° is y. Side adjacent to 60° is 9. Hypotenuse is x.

Angle 60° at bottom right, right angle at top right, so angle at top left is 30°.

Side adjacent to 60° = the vertical leg on the right = 9

In 30-60-90, for 60° angle, adjacent side is the short leg (opposite 30°).

So short leg = 9 = x (in ratio) → then long leg = 9√3 = y (opposite 60°)

Hypotenuse = 2*9 = 18 = x? But x is labeled on the hypotenuse.

In diagram, x is on the hypotenuse, y on the vertical leg? Let's see.

Text: "8) 9 x 60° y" with right angle at top right, so:
- Horizontal leg at top = 9
- Vertical leg at right = y
- Hypotenuse = x
- Angle at bottom right = 60°

For 60° at bottom right:
- adjacent side = vertical leg = y
- opposite side = horizontal leg = 9
- hypotenuse = x

So tan(60°) = opposite/adjacent = 9 / y = √3 → y = 9 / √3 = 3√3

Then hypotenuse x = sqrt(9^2 + (3√3)^2) = sqrt(81 + 27) = sqrt(108) = 6√3

Or using ratios: since opposite 60° = 9 = long leg = x√3 → x=9/√3=3√3 for short leg, but here opposite 60° is 9, so long leg = 9 = x√3 → x=3√3 for short leg, then hypotenuse = 2*3√3 = 6√3

Short leg is adjacent to 60° = y = 3√3

So y = 3√3, x = 6√3

✔ Answer: x = 6√3, y = 3√3

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Problem 9)
Right triangle, 30° at bottom left. Side opposite 30° is y. Side adjacent to 30° is 11√3. Hypotenuse is x.

Angle 30° at bottom left, right angle at top left, so angle at top right is 60°.

Side adjacent to 30° = the vertical leg on the left = 11√3

In 30-60-90, for 30° angle, adjacent side is the long leg (opposite 60°).

So long leg = 11√3 = x√3 → x=11

Short leg = x = 11 = y (opposite 30°)

Hypotenuse = 2x = 22 = x? But x is labeled on the hypotenuse.

In diagram, x is on the hypotenuse, y on the vertical leg? Text: "9) y 11√3 30° x" with right angle at top left, so:
- Vertical leg at left = 11√3
- Horizontal leg at top = y
- Hypotenuse = x
- Angle at bottom left = 30°

For 30° at bottom left:
- adjacent side = vertical leg = 11√3
- opposite side = horizontal leg = y
- hypotenuse = x

So tan(30°) = opposite/adjacent = y / (11√3) = 1/√3 → y = 11√3 / √3 = 11

Then hypotenuse x = sqrt((11√3)^2 + 11^2) = sqrt(363 + 121) = sqrt(484) = 22

Or using ratios: adjacent to 30° = long leg = 11√3 = x√3 → x=11 for short leg, so short leg y = 11, hypotenuse x = 22

So y = 11, x = 22

✔ Answer: x = 22, y = 11

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Problem 10)
Right triangle, 30° at top right. Side opposite 30° is v. Side adjacent to 30° is u. Hypotenuse is 3a? Wait, labeled 3a, but probably 3 times a, but a is not defined. In diagram, it's "3a" on the hypotenuse, but likely it's a typo or meant to be a number. Looking back, in the user input, it's "3a" but in context, probably it's 3 times a variable, but in the image, it might be a number. In the initial description, for problem 10: "10) 3a 30° u v" with right angle at bottom right.

Probably "3a" is meant to be the length, but a is not defined. Perhaps it's 3 times a, but in context, likely it's a constant. In many such problems, it's a number. Perhaps it's 3, and a is part of the label, but unlikely.

Looking at the user's message: "10) 3a 30° u v" and in the image, it might be "3" or "3a". But in the context, probably it's the hypotenuse = 3a, but a is not defined. Perhaps a is a variable, but that doesn't make sense for finding numerical values.

Another possibility: in some fonts, "3a" might be "3" with a subscript, but likely it's 3 times a, but then we can't solve.

Perhaps "3a" is a typo, and it's 3. Or perhaps a is 1, but unlikely.

Let's assume that "3a" means the length is 3a, but then u and v will be in terms of a.

But the instruction is to leave answers as radicals in simplest form, so probably it's a number.

In the initial problem list, for problem 10, it's "3a", but in the image, it might be "3" or "30" etc. Given that in problem 1, hypotenuse is 12, etc., likely "3a" is meant to be a number. Perhaps it's 3, and a is not there.

To resolve, I'll assume that the hypotenuse is 3, as "3a" might be a misread.

But in the user's text, it's "3a", so perhaps a is a variable, but then answers will have a.

Perhaps "3a" is the length, and a is to be ignored, but that doesn't make sense.

Another idea: in some contexts, "a" is used for the short leg, but here it's on the hypotenuse.

Let's read the diagram: "10) 3a 30° u v" with right angle at bottom right, 30° at top right, so angle at bottom left is 60°.

Side labeled 3a is the hypotenuse (from bottom left to top right).

So hypotenuse = 3a

Then for 30° at top right, side opposite 30° = v (horizontal leg at bottom) = short leg = (1/2) * hypotenuse = (1/2)*3a = 3a/2

Side adjacent to 30° = u (vertical leg at right) = long leg = (√3/2) * hypotenuse = (√3/2)*3a = (3a√3)/2

So v = 3a/2, u = (3a√3)/2

But the problem asks for u and v, and if a is a variable, this is fine, but typically in such worksheets, numbers are given.

Perhaps "3a" is a typo, and it's 3. Or perhaps a=1, but unlikely.

In the context of the worksheet, all other problems have numbers, so likely "3a" is meant to be 3.

I'll assume hypotenuse = 3.

Then short leg v = 3/2

Long leg u = (3√3)/2

So u = (3√3)/2, v = 3/2

But to match the format, perhaps keep as fractions.

Since the instruction is to leave as radicals, and 3/2 is fine.

So for problem 10: u = (3√3)/2, v = 3/2

But if "3a" is literal, then u = (3a√3)/2, v = 3a/2

Given that in the answer, it should be in simplest form, and a is not defined, probably it's a number.

Perhaps "3a" is "3" and "a" is part of the label for the side, but in the text, it's "3a" on the hypotenuse.

To be safe, I'll assume that the hypotenuse is 3, as "a" might be a mistake.

So u = (3√3)/2, v = 3/2

✔ Answer: u = \frac{3\sqrt{3}}{2}, v = \frac{3}{2}

But let's confirm with the ratio.

If hypotenuse = 3, then short leg = 3/2, long leg = (3/2)√3 = 3√3/2, yes.

So for problem 10: u = \frac{3\sqrt{3}}{2}, v = \frac{3}{2}

Now, compiling all answers.

Final Answers:

1) n = 6, m = 6√3
2) a = 36√3, b = 36
3) x = 10, y = 5√3
4) x = 39, y = 26√3
5) u = \frac{23\sqrt{3}}{2}, v = \frac{23}{2}
6) n = 6, m = 12
7) a = 10, b = 5
8) x = 6√3, y = 3√3
9) x = 22, y = 11
10) u = \frac{3\sqrt{3}}{2}, v = \frac{3}{2}

For problem 10, if "3a" is meant to be 3a, then u = \frac{3a\sqrt{3}}{2}, v = \frac{3a}{2}, but since the problem likely intends a number, and to match the format, I'll use 3.

In the user's input, it's "3a", but in the context, perhaps a is 1, or it's a typo. Given that, and to provide numerical answers, I'll assume hypotenuse = 3.

So final answer for 10 is u = \frac{3\sqrt{3}}{2}, v = \frac{3}{2}

Now, to box the answers as per the request.

Since the user asked for the final answer, and to provide only the final correct answer after reasoning, I'll list them.

But the instruction is to provide the final answer in a box, but for multiple problems, perhaps list them.

Since it's a worksheet, I'll provide the answers for each.

But in the response format, it says "Final Answer: [Provide only the final correct answer]", so perhaps for the whole thing, but that doesn't make sense.

Perhaps for each problem, but the user uploaded one image with 10 problems.

I think I need to provide the answers for all 10 problems.

So I'll write:

Final Answer:
1) n = 6, m = 6\sqrt{3}
2) a = 36\sqrt{3}, b = 36
3) x = 10, y = 5\sqrt{3}
4) x = 39, y = 26\sqrt{3}
5) u = \frac{23\sqrt{3}}{2}, v = \frac{23}{2}
6) n = 6, m = 12
7) a = 10, b = 5
8) x = 6\sqrt{3}, y = 3\sqrt{3}
9) x = 22, y = 11
10) u = \frac{3\sqrt{3}}{2}, v = \frac{3}{2}

For problem 10, if "3a" is literal, then it should be in terms of a, but I think for the sake of this, I'll keep as above.

To be accurate, let's assume that "3a" means the length is 3a, so answers are in terms of a.

But the problem says "find the missing side lengths", and "leave your answers as radicals in simplest form", so if a is a variable, it's fine.

In that case, for problem 10: hypotenuse = 3a, so short leg v = (1/2)*3a = 3a/2, long leg u = (√3/2)*3a = 3a√3/2

So u = \frac{3a\sqrt{3}}{2}, v = \frac{3a}{2}

And since a is given in the diagram, it's acceptable.

So I'll use that.

Final Answer for 10: u = \frac{3a\sqrt{3}}{2}, v = \frac{3a}{2}

But in the diagram, "3a" is on the hypotenuse, so a is probably a constant, but in math problems, it's common to have variables.

So I'll go with that.

So revised final answers:

1) n = 6, m = 6\sqrt{3}
2) a = 36\sqrt{3}, b = 36
3) x = 10, y = 5\sqrt{3}
4) x = 39, y = 26\sqrt{3}
5) u = \frac{23\sqrt{3}}{2}, v = \frac{23}{2}
6) n = 6, m = 12
7) a = 10, b = 5
8) x = 6\sqrt{3}, y = 3\sqrt{3}
9) x = 22, y = 11
10) u = \frac{3a\sqrt{3}}{2}, v = \frac{3a}{2}

Now for the boxed answer, since it's multiple, I'll list them.

But the instruction is to provide the final answer in a box, so perhaps for the whole set.

Since the user may expect the answers, I'll write them as above.

So Final Answer:
\[ \boxed{\begin{array}{c} \text{1) } n = 6,\ m = 6\sqrt{3} \\ \text{2) } a = 36\sqrt{3},\ b = 36 \\ \text{3) } x = 10,\ y = 5\sqrt{3} \\ \text{4) } x = 39,\ y = 26\sqrt{3} \\ \text{5) } u = \frac{23\sqrt{3}}{2},\ v = \frac{23}{2} \\ \text{6) } n = 6,\ m = 12 \\ \text{7) } a = 10,\ b = 5 \\ \text{8) } x = 6\sqrt{3},\ y = 3\sqrt{3} \\ \text{9) } x = 22,\ y = 11 \\ \text{10) } u = \frac{3a\sqrt{3}}{2},\ v = \frac{3a}{2} \end{array}} \]

For problem 10, if a is not intended, but based on the input, I'll keep it.

To match the format, perhaps the "a" in "3a" is a typo, and it's 3, but I think for accuracy, I'll leave it as is.

So this is the final answer.