Worksheet for calculating side lengths in special right triangles.

You're working on a Special Right Triangles Coloring Activity, which involves solving for missing sides in 45°-45°-90° and 30°-60°-90° triangles using their known side ratios. Let’s go through each problem step-by-step, correct any mistakes, and give you the final answers with explanations.

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## 🔷 Triangle A: 45°-45°-90°

Given: One leg = 12
In a 45°-45°-90° triangle:
- Legs are equal → so other leg (x) = 12
- Hypotenuse = leg × √2 → y = 12√2

✔ Correct!

> x = 12
> y = 12√2
> Color: Blue (as written)

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## 🔷 Triangle B: 45°-45°-90°

Given: Hypotenuse = 18
In a 45°-45°-90° triangle:
- Each leg = hypotenuse / √2 = 18 / √2 = (18√2)/2 = 9√2

So both x and y = 9√2

✘ Your work says “x = ?”, “y = 14” — that’s incorrect.

✔ Correct answer:

> x = 9√2
> y = 9√2
> Color: ??? (not filled — check key on back)

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## 🔷 Triangle C: 30°-60°-90°

Given: Side opposite 30° = 8 (this is the shortest leg)
In a 30°-60°-90° triangle:
- Short leg (opposite 30°) = x = 8? Wait — let’s check the diagram.

Looking at your sketch: angle 30° is at bottom left, right angle at bottom right, so:
- Side adjacent to 30° (horizontal) = x → this is the longer leg
- Side opposite 30° = vertical side = 8 → this is the short leg

Standard ratios:
- Short leg = x/2? No — let’s define properly.

Actually, standard labeling:
- Side opposite 30° = short leg = let’s call it a
- Side opposite 60° = long leg = a√3
- Hypotenuse = 2a

Here, side opposite 30° = 8 → so a = 8

Then:
- Long leg (x) = a√3 = 8√3
- Hypotenuse (y) = 2a = 16

✔ So:

> x = 8√3
> y = 16
> Color: ??? (check key)

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## 🔷 Triangle D: 30°-60°-90°

Given: Hypotenuse = 10
Angle 60° is at bottom left → so side opposite 60° is vertical = x
Side adjacent to 60° (horizontal) = y

In 30°-60°-90°:
- Hypotenuse = 2a → so 2a = 10 → a = 5
- Short leg (opposite 30°) = a = 5 → this is y (adjacent to 60°)
- Long leg (opposite 60°) = a√3 = 5√3 → this is x

✔ So:

> x = 5√3
> y = 5
> Color: ??? (check key)

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## 🔷 Triangle E: 30°-60°-90°

Given: Side opposite 60° = 27 → this is the long leg
So: long leg = a√3 = 27 → a = 27 / √3 = 9√3 (rationalized)

Then:
- Short leg (opposite 30°) = a = 9√3 → this is y
- Hypotenuse = 2a = 18√3 → this is x

✔ So:

> x = 18√3
> y = 9√3
> Color: Dark Purple (as written — looks correct)

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## 🔷 Triangle F: 45°-45°-90°

Given: Hypotenuse = √64 = 8
Each leg = hypotenuse / √2 = 8 / √2 = (8√2)/2 = 4√2

So x and y are both legs → x = y = 4√2

✔ So:

> x = 4√2
> y = 4√2
> Color: ??? (check key)

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## 🔷 Triangle G: 30°-60°-90°

Given: Side adjacent to 30° = 12√3 → this is the long leg (since 30° is at top, adjacent side is horizontal)

So: long leg = a√3 = 12√3 → a = 12

Then:
- Short leg (opposite 30°) = a = 12 → this is x
- Hypotenuse = 2a = 24 → this is y

✔ So:

> x = 12
> y = 24
> Color: ??? (check key)

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## 🔷 Triangle H: 45°-45°-90°

Given: Hypotenuse = 16√5
Each leg = hypotenuse / √2 = (16√5)/√2 = 16√(5/2) = 16√10 / 2 = 8√10

Wait — let’s rationalize properly:

(16√5) / √2 = 16 √(5/2) = 16 * (√10)/2 = 8√10

So both x and y = 8√10

✔ So:

> x = 8√10
> y = 8√10
> Color: ??? (check key)

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## ✔ Final Answers Summary:

| Triangle | x | y | Color |
|----------|----------------|----------------|----------------|
| A | 12 | 12√2 | Blue |
| B | 9√2 | 9√2 | ??? |
| C | 8√3 | 16 | ??? |
| D | 5√3 | 5 | ??? |
| E | 18√3 | 9√3 | Dark Purple |
| F | 4√2 | 4√2 | ??? |
| G | 12 | 24 | ??? |
| H | 8√10 | 8√10 | ??? |

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## 📝 Notes:

- You had some correct answers (A, E), but others were wrong or incomplete (B, C, D, F, G, H).
- Always identify which side is opposite which angle — that determines whether it’s short leg, long leg, or hypotenuse.
- For 45-45-90: legs equal, hypotenuse = leg × √2
- For 30-60-90: short leg : long leg : hypotenuse = 1 : √3 : 2

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## 🎨 Coloring Tip:

Once you have all the numbers, match them to the color key on the back of the worksheet. For example:

- If “12” = Red, then Triangle A gets colored red for x=12
- If “12√2” = Blue, then Triangle A gets colored blue for y=12√2

But since coloring is based on the number (not variable), you may need to assign colors based on the value of x or y — check instructions!

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Let me know if you want me to help you match these numbers to the color key — just provide the key from the back of the page. 😊