This interactive worksheet combines trigonometry practice with creative coloring, helping students master special right triangle ratios through engaging problem-solving.

Let’s solve each problem one by one. These are special right triangles — either 45°-45°-90° or 30°-60°-90°. We’ll use the known side ratios for these triangles.

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Problem 1:
Triangle with angles 45°, 45°, 90°. One leg is $2\sqrt{2}$, and we need to find hypotenuse $a$.

In a 45°-45°-90° triangle:
- Legs are equal.
- Hypotenuse = leg × $\sqrt{2}$

But here, one leg is $2\sqrt{2}$. So:

Hypotenuse $a = 2\sqrt{2} \times \sqrt{2} = 2 \times 2 = 4$

✔ Answer: 4

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Problem 2:
Triangle with angles 45°, 45°, 90°. One leg is 3, and we need to find other leg $y$.

In 45°-45°-90°, legs are equal → so if one leg is 3, the other leg $y = 3$

✔ Answer: 3

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Problem 3:
Triangle with angles 45°, 45°, 90°. Hypotenuse is 6. Find leg $x$.

In 45°-45°-90°:
Leg = hypotenuse ÷ $\sqrt{2}$

So $x = \frac{6}{\sqrt{2}} = \frac{6\sqrt{2}}{2} = 3\sqrt{2}$

Wait — but the directions say “leave your answer in simplest fraction form.” Hmm… $3\sqrt{2}$ is not a fraction. But maybe they mean simplified radical? Let me check the context.

Actually, looking at Problem 6, it has $\frac{10\sqrt{3}}{3}$, which is a fraction with radicals. So perhaps “simplest fraction form” means rationalized denominator if needed, even if it includes radicals.

But in this case, $3\sqrt{2}$ is already simplified. However, let’s double-check: Is there a trig ratio involved?

The problem says: “Find the value of the trigonometric ratios below.” Wait — that might be misleading. Looking again: In each diagram, they label sides (like x, y, a) and ask you to find that side length using trig or special triangle rules.

In Problem 3: Right angle at top left, 45° at top right, hypotenuse = 6, vertical leg = x.

We can use sine or cosine.

sin(45°) = opposite/hypotenuse = x / 6
→ x = 6 × sin(45°) = 6 × $\frac{\sqrt{2}}{2}$ = $3\sqrt{2}$

Same as before.

But the instruction says “simplest fraction form”. Maybe they want it written as a fraction? $3\sqrt{2} = \frac{6\sqrt{2}}{2}$, but that’s not simpler.

Alternatively, perhaps I misread — maybe they just want the numerical coefficient? No, that doesn’t make sense.

Looking ahead to Problem 4: It asks for y, and gives 8 and 60° — likely expecting a fractional answer like $8\sqrt{3}/3$ or something.

Wait — perhaps “trigonometric ratios” refers to sin, cos, tan values? But no, the blanks are labeled with variables like “a = yellow”, meaning find the length of side a.

I think “simplest fraction form” is a bit misleading here — probably they mean simplify any radicals and write as reduced expression. For now, I’ll go with $3\sqrt{2}$, but let’s see if another interpretation fits better.

Alternative approach: Maybe they want the exact value as a fraction involving radicals? Then $3\sqrt{2}$ is fine.

But let’s hold on — perhaps I made a mistake. Let’s look at the triangle orientation.

In Problem 3: Right angle at top left, 45° at top right, so the two legs are horizontal and vertical. The hypotenuse is the diagonal from bottom left to top right, labeled 6. Side x is the vertical leg.

Yes, so x = 6 * sin(45°) = 6 * √2/2 = 3√2.

I think that’s correct.

But wait — the answer format might expect a rational number? No, because later problems have radicals.

Perhaps the “fraction form” applies only when the answer is a rational number? Or maybe it's a generic instruction.

To be safe, I'll compute all and see.

Actually, let’s move to Problem 4 and come back.

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Problem 4:
Right triangle, right angle at top left, 60° at top right, so bottom angle is 30°. Horizontal leg = 8, vertical leg = y, hypotenuse = x.

This is a 30°-60°-90° triangle.

Standard ratios:
- Side opposite 30° : side opposite 60° : hypotenuse = 1 : √3 : 2

Here, the 60° angle is at top right, so the side opposite to it is the vertical leg y.

Side adjacent to 60° is horizontal leg = 8.

In 30-60-90:
- Adjacent to 60° is the shorter leg? No.

Let’s clarify:

If angle at top right is 60°, then:

- Side opposite 60° is the vertical leg y.
- Side adjacent to 60° is the horizontal leg = 8.
- Hypotenuse is x.

In 30-60-90 triangle:
tan(60°) = opposite / adjacent = y / 8
→ y = 8 * tan(60°) = 8 * √3

But that would be y = 8√3, which is large. Let’s check with ratios.

Standard:
- Short leg (opposite 30°) = s
- Long leg (opposite 60°) = s√3
- Hypotenuse = 2s

In this triangle, the 30° angle is at the bottom left (since angles sum to 180: 90+60+30=180).

So side opposite 30° is the horizontal leg? Let's see:

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

Then:
- Side opposite 30° (bottom left) is the top side, which is horizontal leg = 8.
- Side opposite 60° (top right) is the vertical leg = y.
- Hypotenuse is x.

So yes:
Opposite 30° = 8 → so s = 8
Then opposite 60° = s√3 = 8√3 → y = 8√3

But that seems big, and also, typically in such diagrams, the side next to the 60° is the short leg? I'm confusing myself.

Use trig functions directly.

At the 60° angle:
cos(60°) = adjacent / hypotenuse = 8 / x
→ x = 8 / cos(60°) = 8 / (1/2) = 16

sin(60°) = opposite / hypotenuse = y / x = y / 16
→ y = 16 * sin(60°) = 16 * (√3/2) = 8√3

So y = 8√3

But the problem says "simplest fraction form" — 8√3 is not a fraction. Again, same issue.

Perhaps for non-rational answers, they accept radical form.

But let's look at Problem 6: it has 10√3 / 3, which is a fraction.

Maybe in some cases it's fractional.

Another thought: perhaps "trigonometric ratios" means they want sin, cos, tan values, not side lengths? But the blanks are labeled with side names like "a = yellow", so it must be side lengths.

Let's read the directions again: "Find the value of the trigonometric ratios below." But then it says "color the picture according to the color given for the indicated answer", and the answers are associated with side labels.

Perhaps it's poorly worded, and they mean "find the missing side lengths using trigonometry or special triangle properties".

I think that's the case.

For Problem 4, y = 8√3, but let's see if that makes sense with the coloring — but we don't have the coloring key, so we have to trust the math.

But 8√3 is approximately 13.85, while the adjacent side is 8, which is possible.

However, in standard 30-60-90, if the side adjacent to 60° is 8, and 60° is at the top, then the side opposite 30° should be half the hypotenuse.

From above, hypotenuse x = 16, side opposite 30° = 8, which matches (since 8 = 16/2), and side opposite 60° = 8√3, which is correct.

So y = 8√3.

But the instruction says "simplest fraction form" — perhaps they want it as a fraction? 8√3 = \frac{8\sqrt{3}}{1}, but that's silly.

Maybe for this problem, it's acceptable.

Let's do Problem 5 and 6 first.

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Problem 5:
Right triangle, right angle at top right, 60° at bottom right, so top left angle is 30°.

Horizontal leg = 8√3, vertical leg = y, hypotenuse = x.

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

Side opposite 30° is the vertical leg y.
Side opposite 60° is the horizontal leg = 8√3.
Hypotenuse = x.

In 30-60-90:
Opposite 30° : opposite 60° : hypotenuse = 1 : √3 : 2

So, opposite 60° = s√3 = 8√3 → s = 8

Then opposite 30° = s = 8 → y = 8

Hypotenuse = 2s = 16 → x = 16

Using trig:
At 60° angle:
tan(60°) = opposite / adjacent = (8√3) / y
Wait, no: at bottom right 60°, adjacent side is vertical leg y, opposite side is horizontal leg 8√3.

So tan(60°) = opposite / adjacent = (8√3) / y
→ √3 = 8√3 / y
→ y = 8√3 / √3 = 8

Yes, y = 8

Then hypotenuse x: by Pythagoras, x² = y² + (8√3)² = 64 + 64*3 = 64 + 192 = 256 → x = 16

Or sin(60°) = opposite/hypotenuse = 8√3 / x → √3/2 = 8√3 / x → x = 8√3 * 2 / √3 = 16

So x = 16

✔ Answer: 16

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Problem 6:
Right triangle, right angle at bottom left, 30° at bottom right, so top angle is 60°.

Vertical leg = y, horizontal leg = x, hypotenuse = 10√3 / 3

We need to find x.

At 30° angle (bottom right):
adjacent side is horizontal leg x
opposite side is vertical leg y
hypotenuse = 10√3 / 3

cos(30°) = adjacent / hypotenuse = x / (10√3 / 3)

cos(30°) = √3 / 2

So:

√3 / 2 = x / (10√3 / 3)

Solve for x:

x = (√3 / 2) * (10√3 / 3) = (√3 * 10√3) / (2 * 3) = (10 * 3) / 6 = 30 / 6 = 5

Because √3 * √3 = 3

So x = 5

✔ Answer: 5

Now back to Problem 3 and 4.

In Problem 3, we had x = 3√2

In Problem 4, y = 8√3

But the directions say "simplest fraction form". For Problem 6, we got 5, which is integer. For Problem 5, 16. For 1 and 2, 4 and 3.

For Problem 3 and 4, perhaps they expect the answer as is.

But let's double-check Problem 3.

Problem 3: 45-45-90, hypotenuse 6, find leg x.

x = 6 / √2 = 6√2 / 2 = 3√2

Is there a way this could be a fraction? Only if they want decimal, but no, it says simplest fraction form, and 3√2 is irrational.

Perhaps "fraction form" means rationalized, which it is.

Similarly for Problem 4, y = 8√3, which is fine.

But let's look at the coloring hints: for Problem 3, x = orange; Problem 4, y = green. We don't have the key, so we have to assume our calculations are correct.

Another thought: in Problem 4, is the 60° angle positioned such that the side of length 8 is opposite or adjacent?

In the diagram description: "right angle at top left, 60° at top right, horizontal leg = 8, vertical leg = y"

So from the 60° angle at top right, the adjacent side is the horizontal leg (along the top), and the opposite side is the vertical leg down.

Yes, so tan(60°) = opposite/adjacent = y/8, so y = 8 tan(60°) = 8√3

Correct.

Perhaps for "simplest fraction form", they mean if it's a rational number, write as fraction, else leave as radical.

In that case, for Problem 3, 3√2 is fine, but maybe they want it as \frac{3\sqrt{2}}{1}, but that's unnecessary.

I recall that in some contexts, "fraction form" for radicals means to have the radical in numerator, which it is.

So I'll proceed.

But let's list all answers:

1. a = 4
2. y = 3
3. x = 3√2
4. y = 8√3
5. x = 16
6. x = 5

Now, for Problem 3, is there a possibility that x is the other leg? In the diagram, it's labeled as the vertical leg, and since it's 45-45-90, both legs are equal, so if hypotenuse is 6, each leg is 6/√2 = 3√2, yes.

Perhaps the "trigonometric ratio" is sin or cos, but the blank is for the side length.

Another idea: perhaps for each problem, they want the value of a specific trig function, but the labeling suggests otherwise.

Let's read the directions carefully: "Find the value of the trigonometric ratios below." But then it shows diagrams with sides labeled, and blanks with "(a = yellow)", etc.

Perhaps "trigonometric ratios" is a misnomer, and they mean "missing side lengths".

I think that's the case.

To confirm, in Problem 6, we used cos(30°) to find x, and got 5, which is nice.

In Problem 3, if we use sin(45°) = x/6, x = 6 * sin(45°) = 6 * √2/2 = 3√2

Similarly.

Perhaps for Problem 3, they want the answer as a fraction: 3√2 = \frac{6\sqrt{2}}{2}, but that's not simpler.

I think we should go with the simplified radical form.

But let's see Problem 4: y = 8√3

Is there a different interpretation? Suppose the 60° is at the bottom, but the diagram says "60°" at the top right, with right angle at top left.

Perhaps in some conventions, but I think it's correct.

Another way: in 30-60-90, if the side adjacent to 60° is 8, and 60° is acute, then the hypotenuse is 8 / cos(60°) = 8 / 0.5 = 16, and opposite is 16 * sin(60°) = 16 * √3/2 = 8√3, same as before.

So I think it's correct.

Now for the final answers, we need to box them.

But the user will color based on the answer, so we need the numerical value or expression.

For Problem 3, 3√2 is approximately 4.24, but they want exact value.

Similarly for 4, 8√3 ≈ 13.85.

But in the context, since other answers are integers, perhaps I made a mistake in Problem 3 and 4.

Let's reexamine Problem 3.

Diagram: right angle at top left, 45° at top right, so the triangle is oriented with right angle at northwest, 45° at northeast, so the south vertex is the other 45°.

Sides: hypotenuse is from southwest to northeast, labeled 6.

Vertical leg is from northwest to southwest, labeled x.

Horizontal leg is from northwest to northeast, labeled y.

Since it's 45-45-90, legs are equal, so x = y.

By Pythagoras: x^2 + y^2 = 6^2, and x=y, so 2x^2 = 36, x^2 = 18, x = √18 = 3√2

Same as before.

Perhaps the "trigonometric ratio" is for a specific angle, but the blank is for the side.

I think we have to accept that for some problems, the answer is a radical.

But let's look at Problem 2: y = 3, which is integer.

Problem 1: a = 4.

Problem 5: x = 16.

Problem 6: x = 5.

Only 3 and 4 have radicals.

Perhaps in Problem 4, the side of length 8 is the hypotenuse? But the diagram shows it as a leg.

The description says: "horizontal leg = 8", so it's a leg.

Another idea: in Problem 4, if the 60° is at the top, and right angle at top left, then the side between them is the horizontal leg, which is adjacent to 60°, so yes.

Perhaps for "simplest fraction form", they want the answer as a fraction when possible, but for radicals, it's ok.

I recall that in some worksheets, "simplest radical form" is used, and "fraction form" might be a mistake.

Perhaps for Problem 3, they want the value of sin or cos, but the blank is labeled "x = orange", and x is a side.

I think we should proceed with the calculations as is.

So final answers:

1. 4
2. 3
3. 3\sqrt{2}
4. 8\sqrt{3}
5. 16
6. 5

But for Problem 4, let's calculate numerically to verify.

If y = 8√3 ≈ 13.856, and adjacent = 8, then tan(theta) = 13.856/8 = 1.732, which is √3, so theta = 60°, correct.

Similarly for Problem 3, x = 3√2 ≈ 4.24, hypotenuse 6, sin(45°) = 4.24/6 ≈ 0.707, which is √2/2, correct.

So mathematically correct.

Perhaps the "fraction form" applies only to rational answers, and for irrationals, radical form is accepted.

So I'll go with that.

Now for the Final Answer section, since there are six problems, I need to provide all six answers.

The user will use these to color, so we list them.

Final Answer:
1. 4
2. 3
3. \( 3\sqrt{2} \)
4. \( 8\sqrt{3} \)
5. 16
6. 5

But to match the format, perhaps write them clearly.

Since the instruction is to provide the final correct answer after reasoning, and there are multiple parts, I'll list them.

In the response, I can say:

After solving each problem step by step, the answers are:

1. 4
2. 3
3. \( 3\sqrt{2} \)
4. \( 8\sqrt{3} \)
5. 16
6. 5

And box the entire thing or list.

Since it's "Final Answer", perhaps write it as a list.

I think that's fine.

But let's double-check Problem 4 once more.

In Problem 4, is it possible that the side of length 8 is the hypotenuse? The diagram description says "horizontal leg = 8", so no.

Perhaps the 60° is at the bottom, but the text says "60°" at the top right.

Another thought: in some diagrams, the angle might be labeled differently, but based on standard interpretation, it should be correct.

Perhaps for Problem 4, they want y, and in 30-60-90, if the side opposite 30° is 8, then y = 8√3, but if the side adjacent to 60° is 8, and 60° is at the top, then the side opposite 30° is the horizontal leg, which is 8, so yes.

I think it's correct.

So I'll finalize.

One more thing: in Problem 3, the answer is 3√2, but perhaps they want it as a fraction like \frac{3\sqrt{2}}{1}, but that's redundant.

Or maybe simplify to decimal, but no, it says simplest fraction form, and for radicals, it's usually left as is.

I recall that in some contexts, "fraction form" for radicals means to have the radical in the numerator, which it is.

So I think it's fine.

Final Answer:
1. 4
2. 3
3. \( 3\sqrt{2} \)
4. \( 8\sqrt{3} \)
5. 16
6. 5