Solve the right triangle​ ABC, with C+90degrees. A+34.7degrees​, c=17.8 ft. B= ___ degrees ​(Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as​ needed.) a=___ ft ​(Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as​ needed.) b= ___ ft ​(Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as​ needed.)

To solve the right triangle ABC, with \( \angle C = 90^\circ \), \( \angle A = 34.7^\circ \), and \( c = 17.8 \, \text{ft} \) (where \( c \) is the hypotenuse), we need to find:

- \( \angle B \) (the other acute angle),
- side \( a \) (the side opposite angle \( A \)),
- side \( b \) (the side opposite angle \( B \)).

### 1. Find \( \angle B \):
In a right triangle, the sum of the angles is always \( 180^\circ \). Since \( \angle C = 90^\circ \) and \( \angle A = 34.7^\circ \), we can find \( \angle B \) using:
\[
\angle B = 90^\circ - \angle A
\]
\[
\angle B = 90^\circ - 34.7^\circ = 55.3^\circ
\]

### 2. Find side \( a \) (opposite \( \angle A \)):
We can use the sine function:
\[
\sin(A) = \frac{a}{c}
\]
\[
a = c \cdot \sin(A)
\]
Substitute the given values:
\[
a = 17.8 \cdot \sin(34.7^\circ)
\]
\[
a \approx 17.8 \cdot 0.5698 \approx 10.1 \, \text{ft}
\]

### 3. Find side \( b \) (opposite \( \angle B \)):
We can use the cosine function:
\[
\cos(A) = \frac{b}{c}
\]
\[
b = c \cdot \cos(A)
\]
Substitute the values:
\[
b = 17.8 \cdot \cos(34.7^\circ)
\]
\[
b \approx 17.8 \cdot 0.8214 \approx 14.6 \, \text{ft}
\]

### Final Answers:
- \( \angle B = 55.3^\circ \)
- \( a = 10.1 \, \text{ft} \)
- \( b = 14.6 \, \text{ft} \)

Would you like more details or further clarification?

### Related Questions:
1. How can you use the tangent function to find side lengths in a right triangle?
2. What other methods can be used to solve right triangles?
3. How would the solution change if \( \angle A \) was increased?
4. Can you find the area of this right triangle?
5. How would the side lengths change if the hypotenuse was increased?

### Tip:
In any right triangle, once one acute angle is known, you can easily find the other since they always sum to \( 90^\circ \).

Solve the right triangle ABC, with C+90 degrees. A+34.7 degrees, c=17.8 ft. B= ___ degrees (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.) a=___ ft (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.) b= ___ ft (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.)

Learn how to solve a right triangle where angle A = 34.7° and the hypotenuse is c = 17.8 ft. This solution uses trigonometric functions to calculate angle B, side a, and side b, rounding to the nearest tenth. Suitable for students in Grades 9-10.

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