Worksheet on finding missing side lengths in special right triangles, including 45-45-90 and 30-60-90 triangles.

Let’s solve each problem one by one. These are special right triangles — either 45°-45°-90° or 30°-60°-90°. We’ll use their side ratios to find the missing sides.

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Problem 1:
Triangle with legs labeled `a` and `2√2`, angle 45°, so it’s a 45-45-90 triangle.

In a 45-45-90 triangle, the two legs are equal, and hypotenuse = leg × √2.

Here, one leg is 2√2 → so other leg `a` must also be 2√2.

Hypotenuse `b` = leg × √2 = 2√2 × √2 = 2 × 2 = 4

✔ So:
a = 2√2
b = 4

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Problem 2:
Right triangle with angles 45°, 45°, 90°. Hypotenuse = 4.

In 45-45-90: legs = hypotenuse / √2

So each leg (x and y) = 4 / √2 = (4√2)/2 = 2√2

✔ So:
x = 2√2
y = 2√2

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Problem 3:
Right triangle with angle 45°, one leg = (2√2)/2 = √2

Since it’s 45-45-90, both legs are equal → so other leg `f` = √2

Hypotenuse `e` = leg × √2 = √2 × √2 = 2

✔ So:
e = 2
f = √2

Wait — let me double-check:

Given leg = (2√2)/2 = √2 → yes.

Then hypotenuse = √2 * √2 = 2 → correct.

Other leg f = same as given leg = √2 → correct.

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Problem 4:
Same as Problem 3? Leg = 3√2, angle 45° → 45-45-90.

So other leg `f` = 3√2

Hypotenuse `e` = 3√2 × √2 = 3 × 2 = 6

✔ So:
e = 6
f = 3√2

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Problem 5:
Right triangle with angle 45°, hypotenuse = 6 → 45-45-90.

Legs = hypotenuse / √2 = 6 / √2 = (6√2)/2 = 3√2

So x = 3√2, y = 3√2

✔ So:
x = 3√2
y = 3√2

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Problem 6:
Isosceles right triangle (angles 45°, 45°, 90°), hypotenuse = 2√6

Legs = hypotenuse / √2 = 2√6 / √2 = 2√(6/2) = 2√3

So x = 2√3, y = 2√3

✔ So:
x = 2√3
y = 2√3

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Problem 7:
Triangle with angles 30°, 60°, 90°. Side opposite 30° is shortest side.

We’re told side opposite 60° is 16? Wait — look at diagram:

Angle at bottom right is 60°, so side opposite that is the left side (labeled x). But wait — actually, in standard labeling:

If angle at top is 90°, angle at bottom right is 60°, then angle at bottom left is 30°.

Side opposite 30° is the shortest side — which would be the vertical side? But we’re told the base is 16.

Actually, looking again: the side labeled 16 is adjacent to the 60° angle — so it’s the side opposite the 30° angle.

In 30-60-90 triangle:

- Side opposite 30° = short leg = let’s call it `a`
- Side opposite 60° = long leg = a√3
- Hypotenuse = 2a

Here, side opposite 30° is 16 → so a = 16

Then:

- Side opposite 60° (which is x) = a√3 = 16√3
- Hypotenuse (y) = 2a = 32

But wait — check diagram: the side labeled 16 is the base, and angle at bottom right is 60°, so yes — that side is adjacent to 60°, meaning it’s opposite 30°.

So:

Short leg (opposite 30°) = 16 → so a = 16

Long leg (opposite 60°) = x = 16√3

Hypotenuse = y = 32

✔ So:
x = 16√3
y = 32

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Problem 8:
Right triangle with angle 30°, side opposite 30° is 2.

In 30-60-90:

- Opposite 30° = short leg = a = 2
- Opposite 60° = long leg = a√3 = 2√3 → this is v
- Hypotenuse = 2a = 4 → this is u

✔ So:
u = 4
v = 2√3

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Now let’s compile all answers clearly.

Final Answer:

1) a = 2√2, b = 4
2) x = 2√2, y = 2√2
3) e = 2, f = √2
4) e = 6, f = 3√2
5) x = 3√2, y = 3√2
6) x = 2√3, y = 2√3
7) x = 16√3, y = 32
8) u = 4, v = 2√3