Solved Special Triangles (45-45-90 and 30-60-90 ) Worksheet ... - Free Printable

Let’s solve each triangle one by one. We’ll use the special right triangle rules:

- 45°-45°-90° triangles: legs are equal, hypotenuse = leg × √2
- 30°-60°-90° triangles: short leg (opposite 30°) = x, long leg (opposite 60°) = x√3, hypotenuse = 2x

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Problem 1: 45-45-90 triangle

Given: hypotenuse = 14√3
In a 45-45-90 triangle, hypotenuse = leg × √2 → so leg = hypotenuse / √2

So p = q = (14√3) / √2
Rationalize: multiply numerator and denominator by √2 → (14√3 × √2) / 2 = (14√6)/2 = 7√6

✔ p = 7√6, q = 7√6

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Problem 2: 30-60-90 triangle

Angles: 30°, 60°, 90°
Side opposite 60° is given as 6√3 → that’s the “long leg” = x√3
So x√3 = 6√3 → x = 6 (this is the short leg, opposite 30°)

Then:
- y = short leg = 6
- z = hypotenuse = 2x = 12

Wait — let’s check the diagram again. The side labeled 6√3 is adjacent to the 30° angle? Actually, in triangle 2, the right angle is at bottom right, 60° at bottom left, 30° at top. So:

- Side opposite 30° is y (bottom side)
- Side opposite 60° is 6√3 (right side)
- Hypotenuse is z (slanted side)

Yes — so long leg = 6√3 = x√3 → x = 6 → short leg y = 6, hypotenuse z = 12

✔ z = 12, y = 6

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Problem 3: 30-60-90 triangle

Right angle at bottom left. 60° at top, 30° at bottom right.

Side adjacent to 60° (left side) = √5 → this is the short leg? Wait — no.

Actually, side opposite 30° is the shortest side. Here, the side labeled √5 is next to the 60° angle — that means it’s opposite the 30° angle? Let’s think:

Triangle has angles: top=60°, bottom right=30°, bottom left=90°.

So:
- Side opposite 30° (bottom right) is the left vertical side = √5 → that’s the short leg = x
- Then long leg (opposite 60°) = u = x√3 = √5 × √3 = √15
- Hypotenuse v = 2x = 2√5

✔ u = √15, v = 2√5

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Problem 4: 45-45-90 triangle

Hypotenuse s is unknown. Legs r and the other leg (given as 6√2 / 5) are equal? Wait — both base angles are 45°, so yes, it’s isosceles right triangle → legs are equal.

But wait — the side labeled 6√2 / 5 is one leg? And r is the other leg? Since both acute angles are 45°, then r = 6√2 / 5

Then hypotenuse s = leg × √2 = (6√2 / 5) × √2 = (6 × 2) / 5 = 12/5

✔ r = 6√2 / 5, s = 12/5

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Problem 5: 45-45-90 triangle

Hypotenuse = 13√2 / 2
Legs a and b are equal.

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

So a = b = (13√2 / 2) / √2 = 13/2

✔ a = 13/2, b = 13/2

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Problem 6: 30-60-90 triangle

Right angle at top. 30° at bottom left, 60° at bottom right.

Side opposite 30° is h (right side) → short leg = h
Side opposite 60° is 16 (left side) → long leg = 16 = h√3 → so h = 16 / √3 = (16√3)/3

Hypotenuse g = 2h = 2 × (16√3)/3 = (32√3)/3

Wait — let’s double-check labeling.

Diagram: right angle at top vertex. Left side = 16, right side = h, bottom = g.

Angle at bottom left = 30° → so side opposite 30° is h (right side) → yes, short leg = h

Side opposite 60° (at bottom right) is left side = 16 → long leg = 16 = h√3 → h = 16/√3 = (16√3)/3

Hypotenuse g = 2h = (32√3)/3

✔ g = (32√3)/3, h = (16√3)/3

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Problem 7: 30-60-90 triangle

Right angle at bottom left. 60° at top, 30° at bottom right.

Hypotenuse = 18 (side opposite right angle)

In 30-60-90: hypotenuse = 2x → so 2x = 18 → x = 9

Short leg (opposite 30°) = k = x = 9
Long leg (opposite 60°) = j = x√3 = 9√3

✔ j = 9√3, k = 9

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Problem 8: 30-60-90 triangle

Right angle at top left. 30° at bottom left, 60° at top right.

Hypotenuse = 32 (side opposite right angle)

So 2x = 32 → x = 16

Short leg (opposite 30°) = k (top side) = x = 16
Long leg (opposite 60°) = L (left side) = x√3 = 16√3

✔ k = 16, L = 16√3

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Final Answer:

1. p = 7√6, q = 7√6
2. z = 12, y = 6
3. u = √15, v = 2√5
4. r = \frac{6\sqrt{2}}{5}, s = \frac{12}{5}
5. a = \frac{13}{2}, b = \frac{13}{2}
6. g = \frac{32\sqrt{3}}{3}, h = \frac{16\sqrt{3}}{3}
7. j = 9\sqrt{3}, k = 9
8. k = 16, L = 16\sqrt{3}