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The given diagram features two right-angled triangles sharing side \(AC\), and we need to solve for the length of \(AD\). Let's break it down:

### Known information:
1. Triangle \(ABC\) is a right-angled triangle at \(C\).
   - Angle \( \angle ABC = 30^\circ \)
   - Side \(BC = 10\) cm
   - Angle \( \angle BAC = 45^\circ \)

2. Triangle \(ACD\) also shares side \(AC\) and forms a right angle at \(C\). 
   - Side \(CD = 5\) cm

### Steps to solve for \(AD\):

#### 1. Find \(AC\) using triangle \(ABC\):
Since \(ABC\) is a right triangle, we can use trigonometric relationships to find \(AC\). Specifically, use the sine function:

\[
\sin(30^\circ) = \frac{AC}{BC}
\]

\[
AC = BC \cdot \sin(30^\circ) = 10 \cdot 0.5 = 5 \, \text{cm}
\]

#### 2. Find \(AD\) using triangle \(ACD\):
Now that we know \(AC = 5\) cm and \(CD = 5\) cm, triangle \(ACD\) is an isosceles right triangle (since \(\angle ACD = 90^\circ\)). We can use the Pythagorean theorem to find \(AD\):

\[
AD^2 = AC^2 + CD^2
\]

\[
AD^2 = 5^2 + 5^2 = 25 + 25 = 50
\]

\[
AD = \sqrt{50} = 5\sqrt{2} \, \text{cm} \approx 7.07 \, \text{cm}
\]

Thus, the length of \(AD\) is approximately **7.07 cm**.

Let me know if you'd like further clarification or details! Here are some related questions to explore:

1. What other methods can you use to solve for \(AD\)?
2. How would the solution change if \(CD\) was a different length?
3. Can you derive the exact value of \( \sin(45^\circ) \) and \( \sin(30^\circ) \)?
4. How can you verify the result using the Pythagorean theorem?
5. What happens to the angles if triangle \(ACD\) were not isosceles?

**Tip**: Always recheck your intermediate calculations to ensure that the final result is consistent.

Learn how to solve for the length of AD in a geometric figure involving two right triangles, using trigonometric ratios and the Pythagorean theorem. The problem includes key concepts such as sine, angles, and isosceles right triangles. Suitable for Grades 9-11 students.

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