30 60 90 And 45 45 90 Triangle Worksheet - Free Printable

Let’s solve each triangle one by one. These are special right triangles: either 45-45-90 or 30-60-90. We’ll use their side ratios to find x and y.

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Triangle 1 (45-45-90)
Legs are equal, hypotenuse = leg × √2
Given: one leg = 2√2 → so other leg y = 2√2
Hypotenuse x = 2√2 × √2 = 2×2 = 4
→ x = 4, y = 2√2

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Triangle 2 (30-60-90)
Sides: opposite 30° = shortest = a
opposite 60° = a√3
hypotenuse = 2a
Here, hypotenuse = 20 → so 2a = 20 → a = 10
x is opposite 30° → x = 10
y is opposite 60° → y = 10√3
→ x = 10, y = 10√3

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Triangle 3 (30-60-90)
Side opposite 30° = y (shortest)
Side opposite 60° = 3√12 → but let’s simplify: 3√12 = 3×2√3 = 6√3
In 30-60-90: side opposite 60° = y√3 → so y√3 = 6√3 → y = 6
Hypotenuse x = 2y = 12
→ x = 12, y = 6

Wait — check: given side is 3√12. Let me recalculate carefully.

Actually, in Triangle 3:
Angle at bottom left is 60°, angle at top right is 30°, right angle at bottom right.
So side adjacent to 30° (which is vertical) is 3√12 → that’s the side opposite 60°.
Standard ratio:
Opposite 30° : Opposite 60° : Hypotenuse = 1 : √3 : 2
So if opposite 60° = 3√12, then:

Let’s set:
opposite 60° = y√3? No — better to assign:

Let shortest side (opposite 30°) = a
Then opposite 60° = a√3
Hypotenuse = 2a

Here, side labeled 3√12 is opposite 60° → so a√3 = 3√12
Solve for a:
a = 3√12 / √3 = 3 × √(12/3) = 3 × √4 = 3×2 = 6
So a = 6 → which is y (since y is opposite 30°)
Then hypotenuse x = 2a = 12
Yes → x = 12, y = 6

But wait — look at diagram: side labeled 3√12 is vertical, next to 30° angle → actually, it’s adjacent to 30°, so it’s the longer leg → yes, opposite 60° → correct.

Also, simplify 3√12: √12 = 2√3 → 3×2√3 = 6√3 → so opposite 60° = 6√3 → then a√3 = 6√3 → a=6 → same result.

→ x = 12, y = 6

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Triangle 4 (30-60-90)
Top angle 30°, bottom right 60°, right angle at bottom left.
Side opposite 30° is y (vertical)
Side opposite 60° is horizontal = 11√3
Hypotenuse = x

Ratio: opposite 30° : opposite 60° : hyp = 1 : √3 : 2
So opposite 60° = y√3 = 11√3 → y = 11
Then hypotenuse x = 2y = 22
→ x = 22, y = 11

Check: if y=11, then opposite 60° should be 11√3 → matches given → good.

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Triangle 5 (45-45-90)
Both legs equal, hypotenuse = leg × √2
Given hypotenuse = 10 → so leg = 10 / √2 = (10√2)/2 = 5√2
So x = 5√2, y = 5√2
→ x = 5√2, y = 5√2

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Triangle 6 (45-45-90)
Legs are equal → x = y
Given one leg = 6√2 → so both legs = 6√2
Hypotenuse would be 6√2 × √2 = 12, but we don’t need it.
x and y are the two legs → so x = 6√2, y = 6√2
Wait — look at diagram: angle at top is 45°, bottom right 45°, right angle at bottom left.
Side labeled 6√2 is vertical leg → that’s one leg.
x is hypotenuse? Wait no — labels:
Vertical side: 6√2
Horizontal side: y
Hypotenuse: x
Angles: top 45°, bottom right 45° → so legs are equal → y = 6√2
Then hypotenuse x = 6√2 × √2 = 12
Ah! I misread earlier.

So:
Legs: both 6√2 → so y = 6√2
Hypotenuse x = 6√2 * √2 = 6*2 = 12
→ x = 12, y = 6√2

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Triangle 7 (45-45-90)
Isosceles right triangle → legs equal → x = y
Hypotenuse = 8√3
In 45-45-90: hypotenuse = leg × √2 → so leg = hyp / √2 = 8√3 / √2 = 8√(3/2) = 8√6 / 2 = 4√6
Rationalize:
8√3 / √2 = 8√3 * √2 / 2 = 8√6 / 2 = 4√6
So x = 4√6, y = 4√6
→ x = 4√6, y = 4√6

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Triangle 8 (30-60-90)
Angle at top: 60°, bottom right: 30°, right angle at bottom left.
Side opposite 30° is x (vertical)
Side opposite 60° is y (horizontal)
Hypotenuse = 8√5

Ratio: opposite 30° : opposite 60° : hyp = 1 : √3 : 2
So hypotenuse = 2a = 8√5 → a = 4√5
Then x = a = 4√5 (opposite 30°)
y = a√3 = 4√5 * √3 = 4√15
→ x = 4√5, y = 4√15

Double-check:
If x = 4√5, y = 4√15, then hypotenuse = √[(4√5)^2 + (4√15)^2] = √[16*5 + 16*15] = √[80 + 240] = √320 = √(64*5) = 8√5 → correct.

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Now compile all answers:

1. x = 4, y = 2√2
2. x = 10, y = 10√3
3. x = 12, y = 6
4. x = 22, y = 11
5. x = 5√2, y = 5√2
6. x = 12, y = 6√2
7. x = 4√6, y = 4√6
8. x = 4√5, y = 4√15

Final Answer:
1. x = 4, y = 2√2
2. x = 10, y = 10√3
3. x = 12, y = 6
4. x = 22, y = 11
5. x = 5√2, y = 5√2
6. x = 12, y = 6√2
7. x = 4√6, y = 4√6
8. x = 4√5, y = 4√15