https://file.aiquickdraw.com/mathbot/file/1726992900509syvfckg3.jpg

This image features two right triangles: one with a 60° angle and a 45° angle. Let's solve for the missing sides \(d\) and \(b\) using the properties of special right triangles.

### 1. Triangle with 60° and 30° angles:
This triangle has angles of 30°, 60°, and 90°, making it a 30-60-90 triangle. In a 30-60-90 triangle, the sides follow the ratio:
\[
1 : \sqrt{3} : 2
\]
Where:
- The side opposite the 30° angle is \(x\).
- The side opposite the 60° angle is \(x\sqrt{3}\).
- The hypotenuse is \(2x\).

In the given triangle, the side opposite the 60° angle is 2 (which corresponds to \(x\sqrt{3}\)). So, we can solve for \(x\) as follows:
\[
x\sqrt{3} = 2 \quad \Rightarrow \quad x = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
\]
Now, the hypotenuse \(d\) is twice \(x\):
\[
d = 2x = 2 \times \frac{2\sqrt{3}}{3} = \frac{4\sqrt{3}}{3}
\]

### 2. Triangle with 45° angles:
This is a 45-45-90 triangle, where the sides follow the ratio:
\[
1 : 1 : \sqrt{2}
\]
In this case, the legs of the triangle are equal, and the hypotenuse is \(b\sqrt{2}\), where \(b\) is the length of the legs.

The given triangle has a leg length of 6. So, the hypotenuse \(b\) is:
\[
b = \frac{6}{\sqrt{2}} = \frac{6\sqrt{2}}{2} = 3\sqrt{2}
\]

### Final Answers:
- \(d = \frac{4\sqrt{3}}{3}\)
- \(b = 3\sqrt{2}\)

Would you like further details or have any questions?

Here are 5 related questions for more practice:
1. How would the ratio change in a 45°-45°-90° triangle if the hypotenuse is given instead of a leg?
2. What are the exact side lengths for a 30°-60°-90° triangle with a hypotenuse of 10?
3. How can you derive the trigonometric functions for 30° using the 30°-60°-90° triangle?
4. In a 45°-45°-90° triangle, if the hypotenuse is \(7\), what is the length of the legs?
5. How can you simplify expressions like \(\frac{6}{\sqrt{3}}\) using rationalization?

**Tip**: Remember that the sides of 30-60-90 and 45-45-90 triangles always maintain their ratios. Knowing this allows you to quickly find missing sides.

Learn how to find the missing sides d and b in 30°-60°-90° and 45°-45°-90° triangles. Explore key trigonometric properties and solve using simplified radical form. Ideal for students in Grades 9-10.

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