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This problem involves a right-angled triangle where the angle \( B = 90^\circ \), the length of side \( BC = 3 \) cm, and the hypotenuse \( AC = 5 \) cm. We need to find the remaining side and the other angles of the triangle.

### Solution:
Given:
- \( \angle B = 90^\circ \)
- \( BC = 3 \) cm (one of the legs)
- \( AC = 5 \) cm (the hypotenuse)

Using the Pythagorean theorem:
\[
AC^2 = AB^2 + BC^2
\]
Plugging in the known values:
\[
5^2 = AB^2 + 3^2
\]
\[
25 = AB^2 + 9
\]
Subtract 9 from both sides:
\[
16 = AB^2
\]
Now, take the square root of both sides:
\[
AB = 4 \, \text{cm}
\]

### Finding the Angles:
Since it's a right-angled triangle, we already have \( \angle B = 90^\circ \). We now need to find the other two angles, \( \angle A \) and \( \angle C \).

Using trigonometry, we can find these angles. Let's start with \( \angle A \):
\[
\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{3}{5}
\]
\[
A = \arcsin\left(\frac{3}{5}\right) \approx 36.87^\circ
\]

Since the sum of angles in a triangle is \( 180^\circ \):
\[
C = 180^\circ - 90^\circ - 36.87^\circ = 53.13^\circ
\]

### Final Answer:
- The length of side \( AB = 4 \) cm.
- \( \angle A \approx 36.87^\circ \).
- \( \angle C \approx 53.13^\circ \).

Would you like more details or have any questions?

### Further Questions:
1. How would you solve this problem using the cosine or tangent functions?
2. How can the sine rule be applied to find the angles of this triangle?
3. What happens to the other angles if \( BC \) is increased to 4 cm?
4. How does the Pythagorean theorem change if the triangle is not right-angled?
5. Can you use the properties of special right triangles to simplify this problem?

### Tip:
In right-angled triangles, knowing two sides or one side and one angle (other than the right angle) allows you to find all other sides and angles using basic trigonometric functions.

In a right-angle triangle, ∠B = 90°, BC = 3 cm, and AC = 5 cm. Find the length of the remaining side and the angles of the triangle.

Learn how to solve a right-angled triangle problem where ∠B = 90°, BC = 3 cm, and AC = 5 cm. Using the Pythagorean theorem and trigonometric functions, find the length of the missing side and the triangle's angles. Suitable for Grades 7-10.

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