Special Right Triangles Worksheets - Math Monks - Free Printable

Explanation:
We are working with 45°-45°-90° right triangles, which are special because the two legs are equal, and the hypotenuse is leg × √2.

The key relationships (given in the worksheet) are:
- Hypotenuse = Leg × √2
- Leg = Hypotenuse ÷ √2 = Hypotenuse × (√2 / 2)

Let’s solve each triangle one by one:

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1. Right triangle with angles 45°, 45°, 90°. One leg = 8, find hypotenuse *x*.
Since legs are equal, both legs = 8.
Hypotenuse = 8 × √2 → x = 8√2

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2. Right triangle, one leg = 16, angle 45° at the base, so it's a 45-45-90 triangle. The side labeled *x* is the hypotenuse.
Leg = 16 ⇒ Hypotenuse = 16 × √2 → x = 16√2

Wait — check orientation: The right angle is at the top (square mark), and the 45° angle is at bottom left. So the side labeled *x* is opposite the 45° angle — that’s a leg, not the hypotenuse. Let's double-check.

In triangle 2:
- Right angle symbol is at the top vertex.
- One acute angle is 45° at bottom left.
So the side labeled *x* is the side adjacent to the 45° angle and opposite the right angle? No — better to label:

Vertices:
- Right angle at top → vertical and horizontal sides are legs.
- Bottom left angle = 45°, so the side opposite that angle is the vertical leg (from top to bottom right).
But the side labeled *x* is the slanted side (hypotenuse), because it’s opposite the right angle. Yes — the hypotenuse is always opposite the right angle. So *x* is the hypotenuse. The given side (16) is one leg (horizontal). So yes:
Leg = 16 ⇒ Hypotenuse = 16√2 → x = 16√2

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3. Triangle with hypotenuse = 12, right angle at bottom right, 45° at top left. So *x* is the leg (vertical side).
Leg = Hypotenuse ÷ √2 = 12 / √2 = (12√2)/2 = 6√2
→ x = 6√2

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4. Isosceles right triangle: hypotenuse = 24, legs = *x* each.
Leg = 24 / √2 = (24√2)/2 = 12√2
→ x = 12√2

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5. Right triangle, leg = 3√2, 45° angle shown, *x* is the other leg (since it's 45-45-90, legs equal) OR is *x* the hypotenuse? Let's see:
Right angle at bottom left. 45° at top left. So the side labeled *x* is the hypotenuse (opposite right angle). The given side 3√2 is a leg.
So hypotenuse = leg × √2 = 3√2 × √2 = 3 × 2 = 6
→ x = 6

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6. Right triangle, right angle at bottom right, 45° at top right, side labeled 18 is the leg (horizontal), *x* is the other leg (vertical). In 45-45-90, legs are equal → x = 18

Wait — check: If right angle is at bottom right, and 45° at top right, then the vertical side (from top right to bottom right) is one leg, horizontal side (bottom right to bottom left) is the other leg, and slanted side is hypotenuse. The side labeled 18 is the horizontal leg (bottom side), and *x* is the vertical leg (right side). Since both acute angles are 45°, legs are equal → x = 18.

Yes.

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7. Triangle with hypotenuse = 22, 45° angle shown, right angle at bottom left. So *x* is a leg.
Leg = 22 / √2 = (22√2)/2 = 11√2
→ x = 11√2

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8. Square with side = 5, diagonal = *x*. A square’s diagonal splits it into two 45-45-90 triangles. Each triangle has legs = 5, hypotenuse = diagonal = x.
So x = 5√2
→ x = 5√2

Now let’s list all answers clearly:

1. x = 8√2
2. x = 16√2
3. x = 6√2
4. x = 12√2
5. x = 6
6. x = 18
7. x = 11√2
8. x = 5√2

Double-check #5: leg = 3√2, hypotenuse = 3√2 × √2 = 3×2 = 6 ✔
#6: legs equal in 45-45-90 → x = 18 ✔
#8: diagonal of square = side × √2 = 5√2 ✔

All correct.

Final Answer:
1. $8\sqrt{2}$
2. $16\sqrt{2}$
3. $6\sqrt{2}$
4. $12\sqrt{2}$
5. $6$
6. $18$
7. $11\sqrt{2}$
8. $5\sqrt{2}$