Solve the right triangle​ ABC, where C=90___°. a=75.4 ​yd, b=41.8 yd                                                                 c=____yd​(Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as​ needed.)                            A=___° ___′ ​(Simplify your answers. Type integers. Round to the nearest ten minutes as​ needed.)                     B=___° ___′ ​(Simplify your answers. Type integers. Round to the nearest ten minutes as​ needed.)

To solve the right triangle \( \triangle ABC \) where \( C = 90^\circ \), side \( a = 75.4 \) yards, and side \( b = 41.8 \) yards, we need to find:

1. The length of side \( c \) (the hypotenuse).
2. The angle \( A \) in degrees and minutes.
3. The angle \( B \) in degrees and minutes.

### 1. Find the hypotenuse \( c \) using the Pythagorean theorem:
\[
c = \sqrt{a^2 + b^2}
\]
Substituting \( a = 75.4 \) yards and \( b = 41.8 \) yards:
\[
c = \sqrt{75.4^2 + 41.8^2}
\]

Let me calculate this.

\[
c = \sqrt{75.4^2 + 41.8^2} = \sqrt{5680.36 + 1747.24} = \sqrt{7427.6}
\]
\[
c \approx 86.2 \text{ yards}
\]

### 2. Find the angle \( A \):
To find angle \( A \), we use the tangent function:
\[
\tan(A) = \frac{a}{b} = \frac{75.4}{41.8}
\]
\[
A = \tan^{-1} \left( \frac{75.4}{41.8} \right)
\]

Let me calculate this.

\[
A \approx \tan^{-1}(1.8048) \approx 61.5^\circ
\]

Convert the decimal part into minutes:
\[
0.5^\circ \times 60 = 30 \text{ minutes}
\]
Thus, \( A \approx 61^\circ 30' \).

### 3. Find the angle \( B \):
Since \( \triangle ABC \) is a right triangle, the sum of angles is \( 180^\circ \). Therefore, angle \( B \) can be calculated as:
\[
B = 90^\circ - A = 90^\circ - 61^\circ 30'
\]
\[
B \approx 28^\circ 30'
\]

### Final Answer:
- \( c \approx 86.2 \) yards
- \( A \approx 61^\circ 30' \)
- \( B \approx 28^\circ 30' \)

Would you like more details or have any questions?

### Related Questions:
1. What is the sine of angle \( A \)?
2. What is the cosine of angle \( B \)?
3. How would you find the area of triangle \( ABC \)?
4. Can the triangle's area be calculated using trigonometry?
5. What is the ratio of the triangle's sides?

**Tip:** Always remember to round to the required precision after each calculation to avoid errors.

Solve the right triangle ABC, where C = 90°, a = 75.4 yd, b = 41.8 yd, c = ___ yd (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed). A = ___° ___′ (Simplify your answers. Type integers. Round to the nearest ten minutes as needed). B = ___° ___′ (Simplify your answers. Type integers. Round to the nearest ten minutes as needed).

Learn how to solve the right triangle ABC where C = 90°, with side lengths a = 75.4 yd and b = 41.8 yd. This step-by-step guide covers finding the hypotenuse using the Pythagorean Theorem and calculating angles A and B using trigonometric functions. Suitable for high school students in Grades 9-12.

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