Solve the right triangle.
bequals1.23   cequals4.13
.
.
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Question content area top right
Part 1
A
C
B
c
b
a
Question content area bottom
Part 1
Use the values of b and c to find the value of a.
aalmost equals
  

​(Round to the nearest hundredth as ​needed.)
Part 2
Use the values of b and c to find the angle measures of A and B.
Aalmost equals
  
degrees   Balmost equals
  
degrees

To solve the right triangle, we need to find the missing side length \( a \) and the angles \( A \) and \( B \). We are given:

- \( b = 1.23 \)
- \( c = 4.13 \) (hypotenuse)

### Part 1: Finding the Length of Side \( a \)

For a right triangle, the Pythagorean theorem states:

\[
a^2 + b^2 = c^2
\]

Substitute the given values:

\[
a^2 + 1.23^2 = 4.13^2
\]

\[
a^2 + 1.5129 = 17.0569
\]

\[
a^2 = 17.0569 - 1.5129
\]

\[
a^2 = 15.544
\]

\[
a = \sqrt{15.544}
\]

Calculating the value of \( a \):

\[
a \approx 3.94
\]

So, the length of side \( a \) is approximately **3.94**.

### Part 2: Finding the Angles \( A \) and \( B \)

To find the angles, we use trigonometric ratios:

1. **Angle \( A \):**

   \[
   \sin A = \frac{a}{c} = \frac{3.94}{4.13}
   \]

   \[
   \sin A \approx 0.954
   \]

   \[
   A = \arcsin(0.954)
   \]

   Calculating \( A \):

   \[
   A \approx 72.96^\circ
   \]

2. **Angle \( B \):**

   Since the sum of the angles in a triangle is \( 180^\circ \) and one of the angles is \( 90^\circ \), we have:

   \[
   B = 90^\circ - A
   \]

   \[
   B \approx 90^\circ - 72.96^\circ
   \]

   \[
   B \approx 17.04^\circ
   \]

### Final Answers

- Side length \( a \approx 3.94 \)
- Angle \( A \approx 72.96^\circ \)
- Angle \( B \approx 17.04^\circ \)

Would you like more details or have any questions?

### Related Questions:

1. How do you apply the Pythagorean theorem to non-right triangles?
2. What are the formulas for other trigonometric ratios in a right triangle?
3. How can you find the missing side of a triangle using cosine or tangent?
4. What is the relationship between the sine and cosine of complementary angles?
5. How do you solve for the angles in an obtuse triangle?

**Tip:** Remember, in a right triangle, the sum of the two non-right angles will always equal 90 degrees.

Learn how to solve a right triangle with sides b = 1.23 and c = 4.13 using the Pythagorean theorem and trigonometric ratios. Find the length of side a and the angles A and B. Detailed step-by-step solution included. Suitable for high school students.

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