Solve the right triangle using the given information.
aequals=77.1 ​yd, bequals=42.1 yd 
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Part 1

b
a
c
A
B
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Part 1
cequals=enter your response here yd
​(Simplify your answer. Type an integer or a decimal. Round to the nearest tenth if ​needed.)
Aequals=enter your response heredegrees° enter your response hereprime′
​(Simplify your answers. Type integers. Round to the nearest ten minutes if​ needed.)
Bequals=enter your response heredegrees° enter your response hereprime′
​(Simplify your answers. Type integers. Round to the nearest ten minutes if​ needed.)

We are given a right triangle with sides \(a = 77.1\) yd and \(b = 42.1\) yd. Let's solve for the hypotenuse \(c\), and the angles \(A\) and \(B\).

### Step 1: Solve for the hypotenuse \(c\)
The Pythagorean theorem relates the sides of a right triangle:
\[
c = \sqrt{a^2 + b^2}
\]
Substitute \(a = 77.1\) yd and \(b = 42.1\) yd:
\[
c = \sqrt{(77.1)^2 + (42.1)^2}
\]
\[
c = \sqrt{5944.41 + 1772.41} = \sqrt{7716.82}
\]
\[
c \approx 87.8 \, \text{yd}
\]

### Step 2: Solve for angle \(A\)
We can use the tangent function to find angle \(A\) since we know the opposite and adjacent sides:
\[
\tan(A) = \frac{b}{a}
\]
\[
\tan(A) = \frac{42.1}{77.1} \approx 0.546
\]
Now, use the arctangent (inverse tangent) to find \(A\):
\[
A = \tan^{-1}(0.546) \approx 28.6^\circ
\]
Convert the decimal part to minutes by multiplying by 60:
\[
0.6 \times 60 = 36 \, \text{minutes}
\]
So, \(A \approx 28^\circ 36'\).

### Step 3: Solve for angle \(B\)
Since the sum of angles in a triangle is \(180^\circ\) and one angle is \(90^\circ\) (right triangle):
\[
B = 90^\circ - A
\]
\[
B = 90^\circ - 28^\circ 36' = 61^\circ 24'
\]

### Final Answers:
- \(c \approx 87.8\) yd
- \(A \approx 28^\circ 36'\)
- \(B \approx 61^\circ 24'\)

Would you like more details on any part of the solution?

Here are five related questions to expand on this:
1. How would the solution change if we swapped sides \(a\) and \(b\)?
2. What is the area of the triangle given the sides \(a\) and \(b\)?
3. Can we solve for the angles using sine or cosine instead of tangent?
4. How would we solve if only the hypotenuse \(c\) and one side \(a\) were given?
5. What are the applications of trigonometry in real-world problems?

**Tip**: Always keep your calculator in degree mode when working with angles in degrees!

Solve the right triangle using the given information. a = 77.1 yd, b = 42.1 yd. Solve for hypotenuse c, angle A, and angle B.

This page provides a step-by-step solution to solve a right triangle with sides a = 77.1 yd and b = 42.1 yd. Learn how to find the hypotenuse using the Pythagorean Theorem, and calculate angles A and B using trigonometry. Suitable for Grades 9-11.

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