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The problem asks to solve the right triangle \( \triangle ABC \), where \( C = 90^\circ \). The sides given are \( a = 77.7 \) yards and \( b = 48.7 \) yards.

### Steps to Solve:

1. **Find the Hypotenuse (\( c \)):**
   Since it's a right triangle, we use the Pythagorean theorem:
   \[
   c = \sqrt{a^2 + b^2}
   \]
   Substituting the given values:
   \[
   c = \sqrt{(77.7)^2 + (48.7)^2}
   \]

2. **Find Angle \( A \):**
   We use the tangent function:
   \[
   \tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{b}{a}
   \]
   Thus,
   \[
   A = \arctan\left(\frac{b}{a}\right)
   \]

3. **Find Angle \( B \):**
   Since \( A + B = 90^\circ \) (in a right triangle),
   \[
   B = 90^\circ - A
   \]

Now, I'll calculate the values.The solution to the right triangle \( \triangle ABC \) is as follows:

1. **Hypotenuse (\( c \))**: \( c \approx 91.7 \) yards (rounded to the nearest tenth).
2. **Angle \( A \)**: \( A \approx 32.1^\circ \) (rounded to the nearest tenth of a degree).
3. **Angle \( B \)**: \( B \approx 57.9^\circ \) (rounded to the nearest tenth of a degree).

Would you like more details, or do you have any other questions?

Here are 5 related questions to further explore:

1. How would the calculations change if the lengths of \( a \) and \( b \) were in different units, such as feet?
2. What trigonometric functions could be used to find the other sides if only one side and one angle were given?
3. How does the Pythagorean theorem apply to non-right triangles?
4. How would you find the area of this triangle using the given side lengths?
5. Could you solve this triangle using the Law of Sines or the Law of Cosines?

**Tip**: Remember, the Pythagorean theorem only applies to right triangles. For non-right triangles, trigonometric laws like the Law of Sines or Cosines are necessary.

Solve the right triangle ABC, where C = 90°. a = 77.7 yards, b = 48.7 yards. Find c, A, and B.

This right triangle problem uses the Pythagorean theorem to solve for the hypotenuse, and trigonometric functions to find angles A and B. With given side lengths of 77.7 yards and 48.7 yards, find the hypotenuse and angles to the nearest tenth.

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