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Let's analyze the given right triangle \( \triangle ABC \).

We are provided with:
- \( \angle B = 90^\circ \) (right angle),
- \( \angle A = 45^\circ \),
- The length of side \( BC = 49 \) units.

### Step-by-Step Solution:

1. **Find \( AC \) (the hypotenuse):**

Since \( \triangle ABC \) is a 45°-45°-90° triangle, the sides are in a ratio of:
\[
1 : 1 : \sqrt{2}
\]
This means:
\[
AC = BC \times \sqrt{2} = 49 \times \sqrt{2}
\]

2. **Find \( AB \) (the other leg):**

In a 45°-45°-90° triangle, the two legs are equal. Thus:
\[
AB = BC = 49
\]

3. **Find \( \angle C \):**

Since the triangle is a 45°-45°-90° triangle, the other angle \( \angle C \) is also:
\[
m \angle C = 45^\circ
\]

### Final Answers:
\[
AC = 49\sqrt{2}, \quad AB = 49, \quad m\angle C = 45^\circ
\]

Would you like any additional details or clarifications on this solution?

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Here are 5 related questions to expand your understanding:
1. How do you derive the side length ratios in a 45°-45°-90° triangle?
2. Why are both legs of a 45°-45°-90° triangle always equal?
3. How would the solution change if \( \angle A \) were not 45°?
4. Can the Pythagorean theorem be used to verify the results in this problem?
5. What are the characteristics of other special right triangles, such as a 30°-60°-90° triangle?

**Tip:** For any right triangle, knowing two angles automatically determines the third one, since the sum of angles in a triangle is always 180°.

Solve the right triangle given that BC = 49, angle B = 90°, and angle A = 45°. Find the lengths of AC and AB, and the measure of angle C.

Learn how to solve a right triangle problem where BC = 49, with angles B = 90° and A = 45°. This tutorial provides a step-by-step solution for finding AC, AB, and angle C, focusing on the properties of 45°-45°-90° triangles. Suitable for students in Grades 8-10.

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