Special Right Triangles Color by Number activity for math practice.

Okay, now I see the worksheet clearly. The instructions are to find the value of the trigonometric ratios for each triangle and leave the answer in simplest fraction form. Then, I need to color the picture according to the color given for the indicated answer.

The worksheet has six problems, each with a right triangle and an unknown side labeled 'x' or 'y'. The colors corresponding to the answers are:
- (x = yellow)
- (y = purple)
- (x = orange)
- (y = green)
- (x = black)
- (x = blue)

I will start by solving problem 1. It's a 45-45-90 triangle with one leg given as $2\sqrt{2}$. Since it's a special right triangle, the legs are equal, so 'a' should also be $2\sqrt{2}$. To find the hypotenuse 'b', I can use the Pythagorean theorem or the ratio for a 45-45-90 triangle, which is $1:1:\sqrt{2}$. So, if a leg is $2\sqrt{2}$, the hypotenuse is $2\sqrt{2} \times \sqrt{2} = 2 \times 2 = 4$.

Let me now solve problem 2. This is another 45-45-90 triangle, but this time the hypotenuse is given as 3, and I need to find the leg 'y'. Using the same ratio, if the hypotenuse is 3, then each leg is $\frac{3}{\sqrt{2}}$. Rationalizing the denominator, this becomes $\frac{3\sqrt{2}}{2}$.

Now, let's move on to problem 3. This is a 45-45-90 triangle with a leg of 6, and I need to find the other leg 'x'. Again, since the legs are equal, x must also be 6.

For problem 4, I have a 30-60-90 triangle. The side opposite the 30-degree angle is given as 8, and I need to find the side opposite the 60-degree angle, which is 'y'. In a 30-60-90 triangle, the sides are in the ratio $1 : \sqrt{3} : 2$. The side opposite 30° is the shortest side (1 part), and the side opposite 60° is $\sqrt{3}$ times that. So, y = $8\sqrt{3}$.

Problem 5 is also a 30-60-90 triangle. The side opposite the 60-degree angle is given as $8\sqrt{3}$, and I need to find the hypotenuse 'x'. Since the side opposite 60° is $\sqrt{3}$ parts, and the hypotenuse is 2 parts, I can set up a proportion. If $\sqrt{3}$ parts = $8\sqrt{3}$, then 1 part = 8, and 2 parts (the hypotenuse) = 16. So, x = 16.

Finally, for problem 6, I have a 30-60-90 triangle with the side opposite the 30-degree angle given as $10\sqrt{3}$, and I need to find the hypotenuse 'x'. Using the same logic, if 1 part = $10\sqrt{3}$, then 2 parts (the hypotenuse) = $2 \times 10\sqrt{3} = 20\sqrt{3}$.

Now I have all the answers. I will summarize them and explain how to color the mandala.

The solutions are:
1. a = $2\sqrt{2}$, b = 4
2. y = $\frac{3\sqrt{2}}{2}$
3. x = 6
4. y = $8\sqrt{3}$
5. x = 16
6. x = $20\sqrt{3}$

To complete the "Color by Number" activity, you would look at the uncolored mandala and find the sections labeled with these numbers. For example, you would color the section labeled '4' with yellow, the section labeled '$\frac{3\sqrt{2}}{2}$' with purple, the section labeled '6' with orange, the section labeled '$8\sqrt{3}$' with green, the section labeled '16' with black, and the section labeled '$20\sqrt{3}$' with blue. The final colored image is shown on the right side of the original image, demonstrating what the finished product should look like.