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Give angle measures to the nearest degree and side lengths to the nearest tenth please

We are working with a right triangle \( \triangle ABC \), where:

- \( \angle C = 90^\circ \) (since there is a right angle at vertex \( C \)),
- \( \angle A = 50^\circ \) (given),
- \( AC = 5 \) (length of the side opposite \( \angle B \)).

### Step 1: Find \( \angle B \)

Since the sum of the angles in a triangle must be 180°, we can find \( \angle B \) as follows:

\[
\angle B = 180^\circ - 90^\circ - 50^\circ = 40^\circ
\]

### Step 2: Find side \( BC \) (opposite \( \angle A \))

Using the trigonometric relationship for sine:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
For \( \angle A = 50^\circ \):
\[
\sin(50^\circ) = \frac{BC}{AB}
\]
We don't know \( AB \) yet, so we'll use another trigonometric function to first solve for \( AB \).

### Step 3: Find side \( AB \) (hypotenuse)

Using the cosine function for \( \angle A \):
\[
\cos(50^\circ) = \frac{AC}{AB}
\]
Substitute \( AC = 5 \):
\[
\cos(50^\circ) = \frac{5}{AB}
\]
\[
AB = \frac{5}{\cos(50^\circ)} \approx \frac{5}{0.6428} \approx 7.8
\]

### Step 4: Find side \( BC \) (adjacent to \( \angle A \))

Now that we know \( AB \), we can use the sine function from Step 2:
\[
\sin(50^\circ) = \frac{BC}{AB}
\]
Substitute \( AB = 7.8 \):
\[
BC = 7.8 \times \sin(50^\circ) \approx 7.8 \times 0.7660 \approx 6.0
\]

### Final Answers:
- \( BC \approx 6.0 \)
- \( AB \approx 7.8 \)
- \( \angle B = 40^\circ \)

Would you like further explanation or details?

Here are some related questions:
1. How do you use the sine function to find a side length in a triangle?
2. How can the Pythagorean theorem be applied to this triangle?
3. What other trigonometric functions could we have used to solve this problem?
4. How does rounding affect the accuracy of side length calculations?
5. Can this method be applied to non-right triangles?

**Tip:** Always verify your trigonometric calculations with angle measures to avoid errors, especially when rounding!

Learn how to solve the right triangle ABC, given a 50° angle and a side length of 5. This problem involves using trigonometric functions to find the unknown side lengths and angles. Suitable for students in Grades 8-10.

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