Answer key for special right triangles coloring activity, displaying solutions and color codes for each triangle.

You've uploaded an answer key for a "Special Right Triangles Coloring Activity" worksheet. Since this is already the solution sheet, I’ll explain how each problem was solved using the properties of 45°-45°-90° and 30°-60°-90° triangles — the two types of special right triangles.

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## 📐 Special Right Triangle Rules:

1. 45°-45°-90° Triangle

- Legs are equal: `leg = leg`
- Hypotenuse = `leg × √2`
- Ratio: `1 : 1 : √2`

2. 30°-60°-90° Triangle

- Short leg (opposite 30°) = `x`
- Long leg (opposite 60°) = `x√3`
- Hypotenuse (opposite 90°) = `2x`
- Ratio: `1 : √3 : 2`

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# 🔍 Step-by-Step Solutions:

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## 🔹 A. 45°-45°-90° triangle
- One leg = 12 → other leg `x = 12`
- Hypotenuse `y = 12√2`
✔ Answer:
`x = 12`, `y = 12√2` → Color: Dark Blue

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## 🔹 B. 45°-45°-90° triangle
- Hypotenuse = 18 → each leg = `18 / √2 = (18√2)/2 = 9√2`
- So `x = y = 9√2`
✔ Answer:
`x = 9√2`, `y = 9√2` → Color: Yellow

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## 🔹 C. 30°-60°-90° triangle
- Given: short leg (opposite 30°) = 8 → so `x = hypotenuse = 2×8 = 16`
- Long leg `y = 8√3`
✔ Answer:
`x = 16`, `y = 8√3` → Color: Light Purple

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## 🔹 D. 30°-60°-90° triangle
- Given: hypotenuse = 10 → short leg (opposite 30°) = `10/2 = 5` → `y = 5`
- Long leg `x = 5√3`
✔ Answer:
`x = 5√3`, `y = 5` → Color: Red

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## 🔹 E. 30°-60°-90° triangle
- Given: long leg (opposite 60°) = 27 → so `x√3 = 27` → `x = 27/√3 = 9√3`
- Hypotenuse `y = 2x = 18√3`
✔ Answer:
`x = 9√3`, `y = 18√3` → Color: Dark Purple

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## 🔹 F. 45°-45°-90° triangle
- One leg = `√14` → other leg `y = √14`
- Hypotenuse `x = √14 × √2 = √28 = 2√7`
✔ Answer:
`x = 2√7`, `y = √14` → Color: Pink

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## 🔹 G. 30°-60°-90° triangle
- Given: long leg (opposite 60°) = `12√3` → so `x√3 = 12√3` → `x = 12`
Wait — but in the diagram, `x` is the side opposite 30°? Let’s check:

Actually, looking at the triangle:
- Angle 30° is at the top → side opposite it is `x` → that’s the short leg
- Side adjacent to 30° (long leg) = `12√3`
→ So: `short leg = x`, `long leg = x√3 = 12√3` → `x = 12`
→ Then hypotenuse `y = 2x = 24`? But answer says `y = 18`.

⚠️ Wait — let’s double-check.

If `long leg = 12√3`, then `x√3 = 12√3` → `x = 12` → hypotenuse `y = 24`. But answer says `y = 18`.

That doesn’t match. Let’s reverse:

Maybe `x` is the hypotenuse?

In the diagram: angle 30° is at top-left, right angle bottom-right, so side labeled `x` is vertical — opposite the 30° angle? No — actually, side opposite 30° is the horizontal one (bottom), which is not labeled. Labeled sides are:

- `x` = vertical leg (adjacent to 30° → that’s the long leg)
- `y` = hypotenuse
- Given: `x = 12√3` → long leg

So: long leg = `x = 12√3 = short_leg × √3` → short leg = `12`
→ hypotenuse `y = 2 × short leg = 24`

But the answer says `x = 6√3`, `y = 18`

Ah! There’s a mismatch. Let me re-express based on the given answer and work backward.

Given answer: `x = 6√3`, `y = 18`

If `y = 18` (hypotenuse), then short leg = 9, long leg = `9√3`

But they say `x = 6√3` — that would be long leg → then short leg = 6, hypotenuse = 12 — contradiction.

Wait — perhaps the labeling is different.

Looking again at diagram G:

- 30° angle at top-left
- Right angle at bottom-right
- Side labeled `x` is vertical → this is adjacent to 30°, so it’s the long leg
- Side labeled `y` is hypotenuse
- The unlabeled side (bottom) is short leg

Given: `x = 12√3` (long leg) → so short leg = `12√3 / √3 = 12` → hypotenuse `y = 24`

BUT the answer key says `x = 6√3`, `y = 18`

This suggests that maybe the given side is not 12√3 — or there's a typo.

Wait — in the image, the side labeled `12√3` is the hypotenuse? No, it’s labeled on the slanted side? Actually, in the diagram, the side labeled `12√3` is the side opposite the right angle? No — it’s the side between 30° and right angle — that’s the adjacent side to 30°, which is the long leg.

But if we accept the answer key as correct, then:

Given `x = 6√3`, `y = 18`

Then: if `x` is long leg → short leg = `x / √3 = 6√3 / √3 = 6` → hypotenuse = 12 — but `y = 18` contradicts.

Alternatively, if `y = 18` is hypotenuse → short leg = 9, long leg = `9√3`

But answer says `x = 6√3` — not matching.

✔ Conclusion: There appears to be an error in the diagram or answer key for Problem G.

However, since this is an answer key, and you’re asking to solve the problem *as presented*, we must assume the given values are correct and the answers are intended.

Let’s trust the answer key and reverse-engineer:

If `x = 6√3`, `y = 18`, and it’s a 30-60-90 triangle:

Then ratio: short leg : long leg : hypotenuse = `a : a√3 : 2a`

Set `2a = 18` → `a = 9` → long leg = `9√3`

But `x = 6√3` ≠ `9√3` — still inconsistent.

Wait — perhaps `x` is the short leg?

If `x = 6√3` (short leg), then long leg = `6√3 × √3 = 18`, hypotenuse = `12√3`

But diagram shows `12√3` as a side — maybe that’s the hypotenuse?

If hypotenuse = `12√3`, then short leg = `6√3`, long leg = `18`

Then `x = short leg = 6√3`, `y = long leg = 18`

That matches the answer key!

So likely, in diagram G:

- The side labeled `12√3` is the hypotenuse
- `x` is the side opposite 30° → short leg = `6√3`
- `y` is the side opposite 60° → long leg = `18`

✔ So corrected interpretation:

> Given: hypotenuse = `12√3`
> → short leg `x = (12√3)/2 = 6√3`
> → long leg `y = 6√3 × √3 = 18`

✔ Answer:
`x = 6√3`, `y = 18` → Color: Light Green

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## 🔹 H. 45°-45°-90° triangle
- Hypotenuse = `16√5`
- Each leg = `(16√5)/√2 = 16√(5/2) = 16 × √10 / 2 = 8√10`
- So `x = y = 8√10`
✔ Answer:
`x = 8√10`, `y = 8√10` → Color: Light Blue

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# ✔ Final Summary Table:

| Problem | Type | x | y | Number | Color |
|---------|---------------|---------------|---------------|--------|---------------|
| A | 45-45-90 | 12 | 12√2 | 8 | Dark Blue |
| B | 45-45-90 | 9√2 | 9√2 | 3 | Yellow |
| C | 30-60-90 | 16 | 8√3 | 12 | Light Purple |
| D | 30-60-90 | 5√3 | 5 | 5 | Red |
| E | 30-60-90 | 9√3 | 18√3 | 1 | Dark Purple |
| F | 45-45-90 | 2√7 | √14 | 10 | Pink |
| G | 30-60-90 | 6√3 | 18 | 7 | Light Green |
| H | 45-45-90 | 8√10 | 8√10 | 4 | Light Blue |

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# 🎨 Coloring Instructions:

Use the “Number” column to match with a color key (probably on the back of the worksheet). For example:

- Number 8 → Dark Blue → color region A
- Number 3 → Yellow → color region B
- etc.

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✔ You now have a complete explanation of how each value was derived using special right triangle ratios, and how to use the coloring key.

Let me know if you want a printable version or step-by-step diagrams!