In ΔQRS, r = 83 cm, mm∠R=161° and mm∠S=5°. Find the length of q, to the nearest centimeter.
Answer
Attempt 2 out of 2

To find the length of side \( q \) in triangle \( \Delta QRS \), we can use the Law of Sines, which states:

\[
\frac{q}{\sin(\angle R)} = \frac{r}{\sin(\angle S)}
\]

Given:
- \( r = 83 \) cm
- \( \angle R = 161^\circ \)
- \( \angle S = 5^\circ \)

First, we need to find \( \angle Q \), the remaining angle in the triangle:

\[
\angle Q = 180^\circ - \angle R - \angle S = 180^\circ - 161^\circ - 5^\circ = 14^\circ
\]

Now apply the Law of Sines:

\[
\frac{q}{\sin(161^\circ)} = \frac{83}{\sin(5^\circ)}
\]

To find \( q \):

\[
q = \frac{83 \times \sin(161^\circ)}{\sin(5^\circ)}
\]

Now calculate the sine values and solve for \( q \):

\[
q = \frac{83 \times \sin(161^\circ)}{\sin(5^\circ)} \approx \frac{83 \times 0.1736}{0.0872} \approx \frac{14.4128}{0.0872} \approx 165.2 \text{ cm}
\]

To the nearest centimeter:

\[
q \approx 165 \text{ cm}
\]

The length of side \( q \) is approximately **165 cm**.

Would you like more details or have any questions? 

Here are 5 related questions you might consider:
1. How would the Law of Cosines apply if you had all three sides of the triangle?
2. What happens to the sine values as angles approach 0° or 180°?
3. How would the triangle change if \( \angle R \) were slightly less than 161°?
4. What is the significance of knowing the angle sum property of a triangle?
5. How can the Law of Sines be used in non-right triangles?

**Tip:** When using the Law of Sines, always ensure the triangle's angles are consistent with the triangle inequality theorem.

Learn how to find the length of side q in triangle QRS using the Law of Sines. Given r = 83 cm, angle R = 161°, and angle S = 5°, this problem involves applying the Law of Sines to solve for q. Suitable for high school students studying geometry and trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation