Special Right Triangles Coloring Activity worksheet with problems and student work shown.

Let’s solve each special right triangle problem step by step. We’ll use the rules for 45°-45°-90° and 30°-60°-90° triangles.

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Triangle A (45°-45°-90°)
Legs are equal. One leg = 12 → so x = 12
Hypotenuse y = leg × √2 = 12√2

→ x = 12, y = 12√2

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Triangle B (45°-45°-90°)
Base is split into two 9s → total base = 18 → each half = 9
In a 45-45-90, legs are equal → height = 9 → so y = 9
Hypotenuse x = leg × √2 = 9√2

Wait — looking at diagram: it shows an isosceles right triangle with base 18, split down middle → so each half-base = 9, and height = 9 → then hypotenuse (the slanted side) is x = 9√2? But wait — actually, in this case, the two equal sides are the legs, and the base is the hypotenuse? No — let's re-read.

Actually, in Triangle B: It’s drawn as an isosceles right triangle with the right angle at the top, and base 18. The altitude from the right angle splits the base into two 9s. So each leg of the right triangle (from top to bottom corners) is the same length — call that “x”. Then, using Pythagoras on one half:
Each half is a 45-45-90 triangle with legs = 9 and 9 → hypotenuse = 9√2 → but that hypotenuse is actually the leg of the big triangle? Wait no.

Actually, better way: In triangle B, since it’s a 45-45-90 triangle with the right angle at the top, and base 18, then the two legs (left and right sides) are equal. Let’s call each leg “x”. Then by Pythagoras:

x² + x² = 18²
2x² = 324
x² = 162
x = √162 = √(81×2) = 9√2

But wait — the diagram labels the two equal sides as “x” and the base as 18. And there’s a perpendicular dropped from the top to the base, labeled “y”, splitting base into 9 and 9.

So actually, the perpendicular “y” is the height. In an isosceles right triangle, if the hypotenuse is 18, then each leg is 18/√2 = 9√2. But here, the right angle is at the top, so the two legs are the equal sides, and the base is the hypotenuse.

Yes! So base = hypotenuse = 18 → then each leg x = 18 / √2 = 9√2
And the height y (altitude to hypotenuse) in a right triangle = (leg1 × leg2)/hypotenuse = (9√2 × 9√2)/18 = (81 × 2)/18 = 162/18 = 9

Alternatively, in 45-45-90, if hypotenuse is 18, then legs are 18/√2 = 9√2, and altitude to hypotenuse is half the hypotenuse? No — only in isosceles right triangle, altitude to hypotenuse equals half the hypotenuse? Let’s check: area = (1/2)*leg*leg = (1/2)*(9√2)^2 = (1/2)*162 = 81
Also area = (1/2)*base*height = (1/2)*18*y = 9y → so 9y=81 → y=9

So:
x = 9√2
y = 9

But wait — the student wrote x=9, y=18? That can’t be. Let me double-check the diagram description.

Looking back: In triangle B, it says “x” on the two equal sides, “18” on the base, and “y” is the height from top to base. Right angle at top. So yes — base is hypotenuse = 18. Legs are x. So x = 18/√2 = 9√2. Height y = 9.

But the student has written x=9, y=18 — which is swapped and wrong. We must correct.

Actually — perhaps I misread. Maybe the right angle is at the bottom? Diagram says: “right angle symbol at top vertex”? User didn't specify, but typically in such diagrams, if it's labeled with 45° at base angles, then right angle is at top.

To avoid confusion, let’s use standard labeling:

In a 45-45-90 triangle:
- If legs are known, hypotenuse = leg√2
- If hypotenuse is known, leg = hypotenuse/√2

In triangle B: base = 18, and it’s the side opposite the right angle? Or adjacent?

Given the student’s work: they have x=9, y=18 — which suggests they think the legs are 9 and hypotenuse 18 — but that would mean 9√2 ≈12.7 ≠18 — impossible.

Perhaps the 18 is the sum of the two segments? The diagram shows "9" and "9" under the base, so total base=18, and the height is y, and the two slanted sides are x.

Since it’s a 45-45-90, the two base angles are 45°, so the triangle is isosceles right triangle with right angle at top. Therefore, the two legs (from top to left and top to right) are equal — call them L. Then the base (between left and right) is L√2 = 18 → so L = 18/√2 = 9√2. So x = 9√2.

The height y from top to base: in an isosceles right triangle, the altitude to the hypotenuse is half the hypotenuse? Let's calculate: area = (1/2)*L*L = (1/2)*(9√2)^2 = (1/2)*162 = 81. Also area = (1/2)*base*height = (1/2)*18*y = 9y. So 9y=81 → y=9.

So correct values: x = 9√2, y = 9

But the student has x=9, y=18 — which is incorrect. We'll provide correct answer.

However, looking at the student's handwriting: they wrote "x=9", "y=18", and colored something. But mathematically, it should be x=9√2, y=9.

Perhaps the diagram is different? Another possibility: maybe the 18 is not the base but the sum of the two parts? No, it's labeled on the base.

I think we should go with math: for triangle B, with base 18 and 45° at base, right angle at top, then:

- Each leg (x) = 18 / √2 = 9√2
- Height (y) = 9

But let's move on and come back.

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Triangle C (30°-60°-90°)
Short leg (opposite 30°) = 8
Then hypotenuse x = 2 * short leg = 16
Long leg y = short leg * √3 = 8√3

Standard ratios: 30-60-90: sides are 1 : √3 : 2
Opposite 30° is shortest side = 8
So opposite 60° = 8√3 = y
Hypotenuse = 16 = x

→ x = 16, y = 8√3

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Triangle D (30°-60°-90°)
Hypotenuse = 10
Short leg (opposite 30°) = y = hypotenuse / 2 = 5
Long leg (opposite 60°) = x = y * √3 = 5√3

Diagram: right angle at bottom left, 60° at bottom right, so side opposite 60° is x (vertical), side opposite 30° is y (horizontal), hypotenuse=10.

Yes: y = 5, x = 5√3

→ x = 5√3, y = 5

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Triangle E (30°-60°-90°)
Angle at bottom left is 60°, so side opposite 60° is the long leg = 27
In 30-60-90, long leg = short leg * √3
So short leg = long leg / √3 = 27 / √3 = 9√3 (rationalized)
Hypotenuse x = 2 * short leg = 2 * 9√3 = 18√3

Student has x=9√3, y=18√3 — which is swapped.

Let's clarify:
- Angle at bottom left is 60°, so the side opposite to it is the vertical side? Diagram shows: right angle at top, 60° at bottom left, so side opposite 60° is the horizontal side? Need to see.

Typically: in triangle E, it's labeled with 60° at bottom left, right angle at top, so the side between 60° and right angle is adjacent to 60°, etc.

Better: identify which side is which.

Assume:
- Right angle at top vertex.
- 60° at bottom left vertex.
- Then the side opposite 60° is the side from top to bottom right — which is labeled as 27? Student wrote "27" on the side that is not the hypotenuse.

Looking at student's work: they have "27" on the side that is adjacent to 60°? They calculated x=9√3, y=18√3.

Standard approach: in 30-60-90, if you know one side, find others.

Suppose the side labeled 27 is the long leg (opposite 60°). Then:

Long leg = short leg * √3 → short leg = 27 / √3 = 9√3
Hypotenuse = 2 * short leg = 18√3

Now, what is x and y? Diagram: x is probably the hypotenuse, y is the other leg.

Student has x=9√3, y=18√3 — so they might have assigned x to short leg, y to hypotenuse.

But in the diagram, likely x is the hypotenuse, y is the remaining leg.

To match common labeling: let's say in triangle E, the side of length 27 is opposite the 60° angle, so it's the long leg. Then:

Short leg (opposite 30°) = 27 / √3 = 9√3
Hypotenuse = 2 * 9√3 = 18√3

If x is the hypotenuse, then x = 18√3
y is the short leg = 9√3

But student has x=9√3, y=18√3 — so perhaps x is the short leg, y is the hypotenuse? We need to see the diagram labels.

Since the user's image isn't fully clear, but based on standard problems, and the student's answer, perhaps in their diagram, x is the side adjacent or something.

Another way: the student wrote "x=9√3", "y=18√3", and colored dark purple. And they have "CL X=18√3" crossed out, so they changed to x=9√3.

Perhaps the 27 is the hypotenuse? Let's try that.

If hypotenuse = 27, then short leg = 27/2 = 13.5, long leg = 13.5√3 — not matching.

Or if 27 is the short leg, then long leg = 27√3, hypotenuse = 54 — not matching student's answer.

Student has x=9√3, y=18√3, and 27 given. Notice that 9√3 * √3 = 27, so if short leg is 9√3, long leg is 27, hypotenuse 18√3.

Ah! So likely, the side labeled 27 is the long leg (opposite 60°), so short leg = 27 / √3 = 9√3, hypotenuse = 2 * 9√3 = 18√3.

Then if x is the short leg, y is the hypotenuse, then x=9√3, y=18√3 — matches student's answer.

In the diagram, probably x is labeled on the short leg, y on the hypotenuse.

So we'll go with that: x = 9√3, y = 18√3

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Triangle F (45°-45°-90°)
One leg is √44. Simplify √44 = √(4*11) = 2√11
In 45-45-90, both legs equal, so other leg y = 2√11
Hypotenuse x = leg * √2 = 2√11 * √2 = 2√22

Diagram: right angle at bottom left, 45° at bottom right, so legs are vertical and horizontal. Given one leg = √44, so other leg y = √44 = 2√11, hypotenuse x = √44 * √2 = √88 = 2√22

→ x = 2√22, y = 2√11

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Triangle G (30°-60°-90°)
Side labeled 12√3 — which side? Diagram: right angle at bottom right, 30° at top left, so side opposite 30° is the vertical side? Or horizontal?

Typically: if 30° at top left, right angle at bottom right, then the side between them is adjacent to 30°.

Assume: the side of length 12√3 is the long leg (opposite 60°). Because in many problems, they give the long leg.

If long leg = 12√3, then short leg = long leg / √3 = 12√3 / √3 = 12
Hypotenuse = 2 * short leg = 24

Now, what is x and y? Diagram: x is probably the hypotenuse, y is the other leg.

Student hasn't filled, but let's see: if 12√3 is the long leg, then short leg = 12, hypotenuse = 24.

If x is hypotenuse, y is short leg, then x=24, y=12

But which is which? In diagram, likely x is the side opposite 30° or something.

To match: suppose the side labeled 12√3 is opposite the 60° angle, so it's the long leg. Then short leg (opposite 30°) = 12, hypotenuse = 24.

If in the diagram, y is the short leg, x is the hypotenuse, then x=24, y=12

But let's confirm with angles: 30° at top left, so side opposite 30° is the bottom side (horizontal), which might be y or x.

Perhaps it's safer to assume that the given side 12√3 is the long leg, so short leg = 12, hypotenuse = 24.

Then depending on labeling, but typically x and y are the unknowns, so we need to assign.

Looking at the pattern, in previous triangles, x and y are the two unknown sides.

In this case, one side is given as 12√3, and we need to find x and y.

Probably, x is the hypotenuse, y is the other leg.

So if 12√3 is the long leg, then y = short leg = 12, x = hypotenuse = 24

If 12√3 is the short leg, then long leg = 12√3 * √3 = 36, hypotenuse = 24√3 — but that seems large, and student might not have that.

Given that 12√3 is likely the long leg (since it's larger than typical short leg), we'll go with short leg = 12, hypotenuse = 24.

So x = 24, y = 12 (assuming x is hypotenuse, y is short leg)

But to be precise, let's define: in triangle G, with 30° at top left, right angle at bottom right, then:

- Side opposite 30° is the bottom side (let's call it a)
- Side opposite 60° is the left side (b)
- Hypotenuse is the diagonal (c)

Given that one side is 12√3 — if it's b (opposite 60°), then b = 12√3, so a = b / √3 = 12, c = 2a = 24

If the diagram labels x as the hypotenuse, y as the bottom side, then x=24, y=12

We'll assume that.

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Triangle H (45°-45°-90°)
Hypotenuse = 16√5
Then each leg = hypotenuse / √2 = 16√5 / √2 = 16 √(5/2) = 16 √10 / 2 = 8√10? Let's rationalize:

16√5 / √2 = 16 √(5/2) = 16 √10 / 2 = 8√10

Since both legs are equal, x = y = 8√10

Diagram: right angle at bottom, 45° at top right, so legs are the two sides from right angle, hypotenuse is the top side = 16√5

So x and y are the legs, both equal to 8√10

→ x = 8√10, y = 8√10

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Now, let's compile all answers correctly:

A:
x = 12
y = 12√2

B:
After careful thought: in triangle B, with base 18 and 45° at base, right angle at top, then the two legs are equal, and base is hypotenuse = 18.
So each leg x = 18 / √2 = 9√2
Height y = 9 (as calculated)
But in the diagram, y is the height, x is the leg.
So x = 9√2, y = 9

However, the student has x=9, y=18 — which is incorrect. We must provide correct answer.

But let's double-check with another method: in 45-45-90, if the legs are L, hypotenuse L√2. Here, if the two equal sides are x, and base is 18, and it's the hypotenuse, then x√2 = 18 → x = 18/√2 = 9√2. The height to the hypotenuse is (L^2)/(L√2) = L/√2 = 9√2 / √2 = 9. Yes.

So B: x = 9√2, y = 9

C:
x = 16 (hypotenuse)
y = 8√3 (long leg)

D:
x = 5√3 (long leg)
y = 5 (short leg)

E:
Given side 27 is long leg (opposite 60°), so short leg = 27/√3 = 9√3, hypotenuse = 18√3
If x is short leg, y is hypotenuse, then x = 9√3, y = 18√3 — matches student's answer, so we'll keep that.

F:
Leg = √44 = 2√11
Other leg y = 2√11
Hypotenuse x = 2√11 * √2 = 2√22
So x = 2√22, y = 2√11

G:
Assume given side 12√3 is long leg (opposite 60°), so short leg = 12, hypotenuse = 24
If x is hypotenuse, y is short leg, then x = 24, y = 12

H:
Hypotenuse = 16√5
Legs x = y = 16√5 / √2 = 16 √(5/2) = 16 √10 / 2 = 8√10
So x = 8√10, y = 8√10

Now, for the coloring part, we don't have the key, so we'll just provide the numbers.

Final Answers:

A: x = 12, y = 12√2
B: x = 9√2, y = 9
C: x = 16, y = 8√3
D: x = 5√3, y = 5
E: x = 9√3, y = 18√3
F: x = 2√22, y = 2√11
G: x = 24, y = 12
H: x = 8√10, y = 8√10

Note: For triangle B, if the diagram intends differently, but based on standard interpretation, this is correct. Similarly for others.

Let's write the final answer as per the format.