Special Right Triangle - 45°-45°-90° Worksheet for finding side lengths.

Let’s solve each triangle one by one. All of these are 45°-45°-90° right triangles. That means:

✔ The two legs (the sides next to the right angle) are equal in length.
✔ The hypotenuse (side opposite the right angle) is √2 times longer than each leg.

So, if a leg = x, then:
→ other leg = x
→ hypotenuse = x√2

Or, if you know the hypotenuse = h, then:
→ each leg = h / √2 → which we can rationalize to (h√2)/2

We’ll use this for all 8 problems.

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Problem 1:

Triangle has angles 45°, 45°, 90°. Hypotenuse = 12. Legs are x and y — but since it’s 45-45-90, x = y.

So:
x = y = 12 / √2 = (12√2)/2 = 6√2

Wait — let me check the diagram again. In problem 1, side labeled “x” is vertical leg, “y” is horizontal leg, hypotenuse is 12. Since both acute angles are 45°, yes — legs are equal.

So:
x = y = 12 / √2 = 6√2

But sometimes they want decimal? No — keep exact form unless told otherwise.

Actually — wait! Let me double-check labeling.

In problem 1:
Top left corner: 45°, bottom right: 45°, right angle at bottom left.
Side opposite top-left 45° is the bottom leg → that’s y.
Side opposite bottom-right 45° is the left leg → that’s x.
Hypotenuse is 12.

Since both non-right angles are 45°, legs x and y must be equal.

So yes: x = y = 12 / √2 = 6√2

But let me write it as simplified radical: 6√2

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Problem 2:

Legs: one is 8 (bottom), other is x (right side). Hypotenuse is y.

Angles: bottom left 45°, top right 45° → so legs should be equal? Wait — no!

Wait — look: right angle is at bottom right. So legs are bottom (length 8) and right side (length x). Angles: bottom left is 45°, top right is 45° → so yes, it's 45-45-90 → legs equal → x = 8

Then hypotenuse y = 8√2

Yes.

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Problem 3:

Right angle at bottom left. Left leg = 7 (vertical), bottom leg = x, hypotenuse = y.

Angles: top left 45°, bottom right 45° → so legs equal → x = 7

Hypotenuse y = 7√2

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Problem 4:

Same as problem 1 — hypotenuse = 15, legs x and y → equal.

So x = y = 15 / √2 = (15√2)/2

Rationalized: (15√2)/2

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Problem 5:

This one is different — right angle is at the TOP. So legs are the two sides forming the right angle: left side = y, right side = 2√3. Base = x (hypotenuse).

Angles at base: both 45° → so it’s still 45-45-90 → legs equal → y = 2√3

Then hypotenuse x = leg × √2 = 2√3 × √2 = 2√6

Wait — is that right?

Legs are y and 2√3 → since angles at base are both 45°, yes, legs must be equal → y = 2√3

Hypotenuse x = √(leg² + leg²) = √(2 × leg²) = leg√2 = 2√3 × √2 = 2√6

Yes.

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Problem 6:

Right angle at top right. Top side = x (horizontal), right side = y (vertical), hypotenuse = 10√2 (diagonal from bottom left to top right).

Angles: top left 45°, bottom right 45° → so legs x and y are equal.

Hypotenuse = 10√2 → so each leg = (10√2) / √2 = 10

So x = 10, y = 10

Nice and clean.

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Problem 7:

Right angle at top right. Top side = 16 (horizontal), right side = y (vertical), hypotenuse = x (diagonal from bottom left to top right).

Angles: top left 45°, bottom right 45° → so legs equal → y = 16

Hypotenuse x = 16√2

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Problem 8:

Right angle at bottom left. Left leg = √15, bottom leg = y, hypotenuse = x.

Angles: top left 45°, bottom right 45° → so legs equal → y = √15

Hypotenuse x = √15 × √2 = √30

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Now let’s compile answers carefully.

Double-checking each:

1. Hyp=12 → legs = 12/√2 = 6√2 → x=6√2, y=6√2
2. Leg=8 → other leg x=8, hyp y=8√2
3. Leg=7 → other leg x=7, hyp y=7√2
4. Hyp=15 → legs = 15/√2 = (15√2)/2 → x=(15√2)/2, y=(15√2)/2
5. One leg=2√3 → other leg y=2√3, hyp x=2√3 * √2 = 2√6
6. Hyp=10√2 → legs = 10 → x=10, y=10
7. Leg=16 → other leg y=16, hyp x=16√2
8. Leg=√15 → other leg y=√15, hyp x=√15*√2=√30

All seem correct.

Final Answer:

1. x = 6√2, y = 6√2
2. x = 8, y = 8√2
3. x = 7, y = 7√2
4. x = (15√2)/2, y = (15√2)/2
5. x = 2√6, y = 2√3
6. x = 10, y = 10
7. x = 16√2, y = 16
8. x = √30, y = √15