https://file.aiquickdraw.com/mathbot/file/1727220664468zj3x6gjm.jpg

Let's analyze the image and determine how to calculate the value of \( \cos(q) \).

The diagram consists of two right triangles, \( \triangle ABC \) and \( \triangle ADE \). From the given side lengths:

- In \( \triangle ABC \):
  - \( AB = \sqrt{5} \)
  - \( BC = 1 \)
  - \( AC = 2 \)
  
We can confirm that \( \triangle ABC \) is a right triangle by using the Pythagorean theorem:
\[
AB^2 = AC^2 + BC^2 \quad \Rightarrow \quad (\sqrt{5})^2 = 2^2 + 1^2 \quad \Rightarrow \quad 5 = 4 + 1.
\]
This confirms the relationship.

### Calculating \( \cos(q) \):
- \( q \) is the angle at \( A \) in \( \triangle ABC \).
- By definition of cosine in a right triangle:
\[
\cos(q) = \frac{\text{adjacent}}{\text{hypotenuse}}.
\]
In \( \triangle ABC \), the side adjacent to \( q \) is \( AC \) (which equals 2), and the hypotenuse is \( AB \) (which equals \( \sqrt{5} \)).

Thus:
\[
\cos(q) = \frac{AC}{AB} = \frac{2}{\sqrt{5}}.
\]

You can rationalize the denominator if needed:
\[
\cos(q) = \frac{2}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{5}.
\]

So, the value of \( \cos(q) \) is \( \frac{2\sqrt{5}}{5} \).

---

Would you like a further breakdown of any part of this? Let me know if you have questions.

Here are some related questions to deepen your understanding:
1. What is the sine of angle \( q \) in \( \triangle ABC \)?
2. How would you find the tangent of angle \( q \)?
3. What is the area of \( \triangle ABC \)?
4. How can you apply the Pythagorean theorem in triangle \( ADE \)?
5. How do you calculate the cosine of angle \( p \) in \( \triangle ADE \)?

**Tip:** Always ensure the correct identification of adjacent and opposite sides relative to the angle when using trigonometric ratios in right triangles.

Determine the value of cos(q) for the given right triangles.

Learn how to determine the value of cos(q) for right triangle ABC using trigonometric ratios. This problem applies the cosine function formula, the Pythagorean theorem, and covers right triangle properties. Suitable for high school students in Grades 9-10.

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