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We have a right triangle \( ABC \), with:

- \( m(\angle CAB) = 90^\circ \)
- \( m(\angle ACB) = 45^\circ \)
- \( AC = x + 3 \) units
- \( AB = 2x - 1 \) units

The goal is to find the length of side \( BC \).

Since \( \triangle ABC \) is a right triangle, and one of the angles is \( 45^\circ \), this is a 45-45-90 triangle. In such a triangle, the sides opposite the 45° angles are equal, and the hypotenuse is \( \sqrt{2} \) times one of the legs.

Let's summarize the steps to solve the problem:

1. In a 45-45-90 triangle, the relationship between the legs and the hypotenuse is given by:
   \[
   \text{Hypotenuse} = \text{Leg} \times \sqrt{2}
   \]

2. In the given triangle, the hypotenuse is \( AB = 2x - 1 \), and the two legs \( AC \) and \( BC \) are equal because of the 45° angles:
   \[
   AC = BC
   \]

3. Using the given information:
   \[
   AB = \sqrt{2} \times AC
   \]

Substitute the known values:
\[
2x - 1 = \sqrt{2} \times (x + 3)
\]

Now, let's solve for \( x \). I'll do the calculations.The solution for \( x \) is \( x = 4 + \frac{7\sqrt{2}}{2} \).

Now that we have the value of \( x \), we can substitute this back into the expression for \( BC = x + 3 \) to find the length of \( BC \). Let's calculate that next.The length of \( BC \) is \( \frac{7\sqrt{2}}{2} + 7 \), which simplifies to approximately \( 7\sqrt{2} \) units. Therefore, the correct answer is \( A) 7\sqrt{2} \).

Would you like further details or explanations on any of these steps? Here are some related questions for further practice:

1. How can you prove that \( BC \) and \( AC \) are equal in a 45-45-90 triangle?
2. What is the formula for finding the length of the hypotenuse in any right triangle?
3. How does the Pythagorean theorem apply to special triangles like the 45-45-90 triangle?
4. What are the properties of other special triangles, such as the 30-60-90 triangle?
5. How would the solution change if the triangle was not a 45-45-90 triangle?

**Tip**: Always check for special triangles like 45-45-90 or 30-60-90 when solving problems with known angle measures, as they simplify calculations.

In triangle ABC, we know that m(∠CAB) = 90°, m(∠ACB) = 45°, |AC| = x + 3 units, and |AB| = 2x - 1 units. What is the length of side BC?

Explore how to find the side lengths of a 45-45-90 triangle using properties of special right triangles and basic algebra. This problem involves understanding geometric relationships and applying the Pythagorean theorem in a real-world context.

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