Grade 8 Special Right Triangles Worksheets 2024 - Free Printable

You’ve uploaded a worksheet titled “Special Right Triangles” from Kuta Software — Infinite Geometry. The goal is to find the missing side lengths in 8 special right triangles and leave answers as radicals in simplest form.

The two types of special right triangles featured are:
- 45°–45°–90° triangles (isosceles right triangles)
- 30°–60°–90° triangles

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## 🔑 Key Ratios for Special Right Triangles

✔ 45°–45°–90° Triangle

Sides are in the ratio:
1 : 1 : √2
→ Legs are equal; hypotenuse = leg × √2

✔ 30°–60°–90° Triangle

Sides are in the ratio:
1 : √3 : 2
→ Short leg (opposite 30°) : Long leg (opposite 60°) : Hypotenuse
So:
- Short leg = x
- Long leg = x√3
- Hypotenuse = 2x

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Now, let’s go problem by problem and explain the solution.

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## 📐 Problem 1:

Triangle with:
- One leg = `2√2`
- Angles: 45°, 90° → so it’s a 45-45-90 triangle
- Missing: hypotenuse `a`, other leg `b`

✔ Since it’s 45-45-90, legs are equal → `b = 2√2`

✔ Hypotenuse = leg × √2 = `2√2 × √2 = 2 × 2 = 4` → `a = 4`

✔️ Answer: a = 4, b = 2√2

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## 📐 Problem 2:

Triangle with:
- Hypotenuse = 4
- Angle = 45° → 45-45-90 triangle
- Missing: legs `x` and `y`

✔ In 45-45-90, each leg = hypotenuse / √2 = `4 / √2`

Rationalize: `(4 / √2) × (√2/√2) = (4√2)/2 = 2√2`

✔️ Answer: x = 2√2, y = 2√2

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## 📐 Problem 3:

Given leg = `3√2 / 2`, angle 45° → 45-45-90

✔ Other leg `y = 3√2 / 2` (equal legs)

✔ Hypotenuse `x = leg × √2 = (3√2 / 2) × √2 = (3 × 2) / 2 = 3`

✔️ Answer: x = 3, y = 3√2 / 2

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## 📐 Problem 4:

Given leg = `3√2`, angle 45° → 45-45-90

✔ Other leg `y = 3√2`

✔ Hypotenuse `x = 3√2 × √2 = 3 × 2 = 6`

✔️ Answer: x = 6, y = 3√2

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## 📐 Problem 5:

Hypotenuse = 6, angle 45° → 45-45-90

✔ Each leg = `6 / √2 = (6√2)/2 = 3√2`

✔️ Answer: x = 3√2, y = 3√2

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## 📐 Problem 6:

This is an isosceles right triangle (45° at base, right angle at top), hypotenuse = `2√6`

✔ Legs are equal → let each leg be `x` and `y`

In 45-45-90: hypotenuse = leg × √2 → leg = hypotenuse / √2

→ `x = y = (2√6) / √2 = 2√(6/2) = 2√3`

✔️ Answer: x = 2√3, y = 2√3

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## 📐 Problem 7:

Triangle with angles: 45°, 60°, and 90°? Wait — sum must be 180°.

Wait — 45° + 60° = 105° → so third angle = 75°? But that’s not a standard special triangle.

But look — the diagram shows a right angle, a 60° angle, and the third angle must be 30° — but it’s labeled 45°? That’s a mistake.

Actually, looking at the diagram again — it’s likely a 30-60-90 triangle with the 60° angle shown, right angle, and the unlabeled angle is 30°. The 45° label might be a typo or misplacement.

But given the answer provided is `x = 8√3, y = 8`, and hypotenuse = 16, let’s verify.

If hypotenuse = 16 → in 30-60-90:
- Short leg (opposite 30°) = 16/2 = 8 → that’s `y`
- Long leg (opposite 60°) = 8√3 → that’s `x`

So the 45° label is probably incorrect — this is a 30-60-90 triangle.

✔ So:
- Side opposite 30° = y = 8
- Side opposite 60° = x = 8√3
- Hypotenuse = 16

✔️ Answer: x = 8√3, y = 8

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## 📐 Problem 8:

Given:
- One leg = 2 (opposite 30° — since it’s adjacent to 60° angle)
- Angle = 30°, right angle → so it’s a 30-60-90 triangle

✔ Short leg (opposite 30°) = 2 → so hypotenuse `u = 2 × 2 = 4`

✔ Long leg (opposite 60°) = 2 × √3 = `2√3` → that’s `v`

✔️ Answer: u = 4, v = 2√3

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## ✔ Final Answers (as per worksheet):

1. a = 4, b = 2√2
2. x = 2√2, y = 2√2
3. x = 3, y = 3√2 / 2
4. x = 6, y = 3√2
5. x = 3√2, y = 3√2
6. x = 2√3, y = 2√3
7. x = 8√3, y = 8 *(Note: likely a 30-60-90 triangle despite 45° label)*
8. u = 4, v = 2√3

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## 💡 Summary:

All problems use the ratios of 45-45-90 or 30-60-90 triangles. Always identify which type you have based on angles, then apply the side ratios. Remember to rationalize denominators when needed.

Let me know if you want step-by-step diagrams or more practice!