https://file.aiquickdraw.com/mathbot/file/17269948401267cgem72x.jpg

The image shows two right triangles, each with specific angle measurements and one missing side labeled for calculation.

### Triangle 1 (left):
- Angles: \( 30^\circ, 60^\circ, 90^\circ \)
- Hypotenuse: \( 5 \)
- Unknown side \( d \) (adjacent to the \( 30^\circ \) angle)

### Triangle 2 (right):
- Angles: \( 45^\circ, 45^\circ, 90^\circ \)
- Given leg: \( 4 \) (leg adjacent to both \( 45^\circ \) angles)
- Unknown side \( a \) (other leg, also adjacent to \( 45^\circ \))

Now let's solve for \( d \) and \( a \).

#### 1. Solving for \( d \) in Triangle 1 (30-60-90 triangle):
In a 30-60-90 triangle, the side lengths follow the ratio:
\[
1 : \sqrt{3} : 2
\]
Here, the hypotenuse corresponds to \( 2 \), and the side opposite \( 30^\circ \) corresponds to \( 1 \), so:
\[
\text{Adjacent side} = \frac{\text{Hypotenuse}}{2} = \frac{5}{2} = 2.5
\]
\[
d = 5 \cdot \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2}
\]

#### 2. Solving for \( a \) in Triangle 2 (45-45-90 triangle):
In a 45-45-90 triangle, both legs are equal, so:
\[
a = 4
\]

Now I'll compute \( d \approx \frac{5\sqrt{3}}{2} \).For the given triangles:
- The length of side \( d \) is approximately \( 4.33 \).
- The length of side \( a \) is \( 4 \).

Would you like further details or explanations? Here are 5 related questions to deepen your understanding:

1. How do the side ratios work in special right triangles like 30-60-90?
2. Why do the legs of a 45-45-90 triangle have the same length?
3. How can the Pythagorean theorem be used to verify these side lengths?
4. What is the geometric significance of the square root of 3 in the 30-60-90 triangle?
5. How are these triangle properties applied in real-world scenarios like engineering?

**Tip:** For any right triangle, trigonometric functions (like sine and cosine) are powerful tools to calculate unknown sides when angles are known.

The image contains two right triangles. The first triangle has angles of 30°, 60°, and 90°, with a hypotenuse of 5. The second triangle has angles of 45°, 45°, and 90°, with one leg measuring 4. The goal is to find the lengths of the unknown sides labeled as d and a.

Learn how to calculate the unknown sides of two special right triangles: a 30-60-90 triangle with a hypotenuse of 5, and a 45-45-90 triangle with one leg measuring 4. This solution explores key trigonometric concepts and side ratios used in solving these problems.

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