Special Right Triangles Color by Number - Free Printable

Let's solve each of the six problems involving special right triangles (45°-45°-90° and 30°-60°-90° triangles). We'll use the known ratios for these special triangles and apply trigonometric principles or the Pythagorean Theorem where necessary. Then we’ll determine the missing side values.

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🔷 Special Right Triangle Ratios:

1. 45°-45°-90° triangle:
- Legs are equal: $ a = b $
- Hypotenuse: $ a\sqrt{2} $

2. 30°-60°-90° triangle:
- Side opposite 30°: $ x $
- Side opposite 60°: $ x\sqrt{3} $
- Hypotenuse: $ 2x $

We will now go through each problem.

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✔ Problem 1:

Triangle: 45°-45°-90°, one leg is $ 2\sqrt{2} $, find $ a $ (hypotenuse).

Since it’s a 45°-45°-90° triangle, both legs are equal. So the other leg $ b = 2\sqrt{2} $.
Hypotenuse $ a = \text{leg} \times \sqrt{2} = 2\sqrt{2} \times \sqrt{2} = 2 \times 2 = 4 $

✔ Answer: $ a = 4 $ → Color: Yellow

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✔ Problem 2:

Triangle: 45°-45°-90°, one leg is 3, find $ y $ (other leg)

In a 45°-45°-90° triangle, legs are equal. So $ y = 3 $

✔ Answer: $ y = 3 $ → Color: Purple

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✔ Problem 3:

Triangle: 45°-45°-90°, hypotenuse is 6, find $ x $ (one leg)

In 45°-45°-90°, hypotenuse = $ \text{leg} \times \sqrt{2} $
So:
$ x\sqrt{2} = 6 $
$ x = \frac{6}{\sqrt{2}} = \frac{6\sqrt{2}}{2} = 3\sqrt{2} $

✔ Answer: $ x = 3\sqrt{2} $ → Color: Orange

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✔ Problem 4:

Triangle: 30°-60°-90°, side adjacent to 60° is 8 (this is the shorter leg), find $ y $

Let’s identify the sides:

- Angle at top is 60°
- Right angle on left
- So the side of length 8 is opposite the 30° angle, meaning it's the shorter leg.

In a 30°-60°-90° triangle:
- Short leg (opposite 30°) = $ x $
- Long leg (opposite 60°) = $ x\sqrt{3} $
- Hypotenuse = $ 2x $

Here, short leg = 8 ⇒ $ x = 8 $

Then long leg $ y = x\sqrt{3} = 8\sqrt{3} $

✔ Answer: $ y = 8\sqrt{3} $ → Color: Green

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✔ Problem 5:

Triangle: 30°-60°-90°, side opposite 60° is $ 8\sqrt{3} $, find $ x $ (hypotenuse)

Given:
- Side opposite 60° = $ 8\sqrt{3} $
- In 30°-60°-90° triangle: opposite 60° = $ x\sqrt{3} $
- So: $ x\sqrt{3} = 8\sqrt{3} $ ⇒ $ x = 8 $

But wait — this $ x $ is the shorter leg (opposite 30°)? No!

Wait: Let’s clarify.

Actually, in a 30°-60°-90° triangle:
- Side opposite 30°: $ x $
- Side opposite 60°: $ x\sqrt{3} $
- Hypotenuse: $ 2x $

We're told that the side opposite 60° is $ 8\sqrt{3} $, so:
$$
x\sqrt{3} = 8\sqrt{3} \Rightarrow x = 8
$$

So:
- Short leg (opposite 30°) = 8
- Hypotenuse = $ 2x = 16 $

But the question asks for $ x $, which is labeled as the hypotenuse in the diagram.

Wait! Look at the diagram:

- The triangle has:
- A right angle at the top-right corner.
- A 60° angle at the bottom-right.
- So the side labeled $ x $ is the hypotenuse.

Yes — $ x $ is the hypotenuse.

So if the side opposite 60° is $ 8\sqrt{3} $, then:
- That side = $ x\sqrt{3} $? Wait — no.

Let’s define:
- Let $ s $ be the side opposite 30°
- Then side opposite 60° = $ s\sqrt{3} $
- Hypotenuse = $ 2s $

We’re given: side opposite 60° = $ 8\sqrt{3} $

So:
$$
s\sqrt{3} = 8\sqrt{3} \Rightarrow s = 8
$$

Then hypotenuse $ x = 2s = 16 $

✔ Answer: $ x = 16 $ → Color: Black

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✔ Problem 6:

Triangle: 30°-60°-90°, hypotenuse is $ \frac{10\sqrt{3}}{3} $, find $ x $ (side adjacent to 30°, i.e., longer leg?)

Let’s analyze:

- Right angle at bottom-left
- 30° angle at bottom-right
- So hypotenuse is the side opposite the right angle: $ \frac{10\sqrt{3}}{3} $
- Side $ x $ is adjacent to 30°, and lies along the base — this is the longer leg (opposite 60°)
- Side $ y $ is vertical leg, opposite 30° → shorter leg

In a 30°-60°-90° triangle:
- Hypotenuse = $ 2x $ (where $ x $ is the short leg)
- But here, let’s use standard variables.

Let:
- Short leg (opposite 30°) = $ y $
- Long leg (opposite 60°) = $ x $
- Hypotenuse = $ 2y $

But hypotenuse is given as $ \frac{10\sqrt{3}}{3} $, so:
$$
2y = \frac{10\sqrt{3}}{3} \Rightarrow y = \frac{5\sqrt{3}}{3}
$$

Now, long leg $ x = y\sqrt{3} = \frac{5\sqrt{3}}{3} \cdot \sqrt{3} = \frac{5 \cdot 3}{3} = 5 $

✔ Answer: $ x = 5 $ → Color: Blue

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✔ Final Answers:

| Problem | Answer | Color |
|--------|--------------|-----------|
| 1 | $ a = 4 $ | Yellow |
| 2 | $ y = 3 $ | Purple |
| 3 | $ x = 3\sqrt{2} $ | Orange |
| 4 | $ y = 8\sqrt{3} $ | Green |
| 5 | $ x = 16 $ | Black |
| 6 | $ x = 5 $ | Blue |

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🎨 How to Color:

- Use the color key provided:
- $ a = 4 $ → Yellow
- $ y = 3 $ → Purple
- $ x = 3\sqrt{2} $ → Orange
- $ y = 8\sqrt{3} $ → Green
- $ x = 16 $ → Black
- $ x = 5 $ → Blue

Color the corresponding sections of the picture based on these answers.

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✔ Summary of Work:

All problems were solved using properties of special right triangles:
- 45°-45°-90°: legs equal, hypotenuse = leg × √2
- 30°-60°-90°: sides in ratio $ 1 : \sqrt{3} : 2 $

No trigonometry needed — just geometric ratios.

Let me know if you'd like a visual explanation or a printable version!