https://file.aiquickdraw.com/mathbot/file/17270496579043d7wsbl6.jpg

Let's solve the right triangle problem from your image.

### Given:
- \( C = 90^\circ \) (Right angle)
- \( a = 19.1 \, \text{cm} \) (Opposite side)
- \( c = 45.7 \, \text{cm} \) (Hypotenuse)
- Need to find \( b \), \( A \), and \( B \).

#### Step 1: Find side \( b \) using the Pythagorean theorem
\[
c^2 = a^2 + b^2
\]
\[
b = \sqrt{c^2 - a^2}
\]
Substitute the values:
\[
b = \sqrt{45.7^2 - 19.1^2}
\]

#### Step 2: Find angle \( A \) using the sine function
\[
\sin(A) = \frac{a}{c}
\]
\[
A = \sin^{-1}\left(\frac{19.1}{45.7}\right)
\]

#### Step 3: Find angle \( B \) using the sum of angles in a triangle
\[
B = 90^\circ - A
\]

Let me calculate these for you.### Solutions:

- \( b \approx 41.52 \, \text{cm} \) (rounded to the nearest hundredth)
- \( A \approx 24^\circ 42' \) (rounded to the nearest minute)
- \( B \approx 65^\circ 17' \) (rounded to the nearest minute)

Would you like more details or have any questions?

Here are 5 related questions to extend your learning:
1. How would the calculation change if side \( a \) were doubled?
2. Can you solve the same triangle using cosine instead of sine?
3. What is the relationship between angles \( A \) and \( B \) in any right triangle?
4. How does the accuracy of your answer change if the values for \( a \) or \( c \) were given with more decimal places?
5. What are some common errors when applying the Pythagorean theorem?

**Tip:** Always double-check the accuracy of your angles by verifying that the sum of all angles in a triangle equals 180°.

Solve the right triangle ABC, where C = 90°. Given: a = 19.1 cm, c = 45.7 cm. Find side b and angles A and B.

Learn how to solve right triangle ABC where C = 90°, a = 19.1 cm, and c = 45.7 cm. Find side b using the Pythagorean theorem and calculate angles A and B using trigonometric ratios. Detailed step-by-step solution for high school students.

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