Given triangle , which of the following sets of side lengths could represent the sides of a triangle that satisfies the Law of Sines?
Table of contents
- 0. Review of College Algebra4h 45m
- 1. Measuring Angles40m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
7. Non-Right Triangles
Law of Sines
Multiple Choice
Given triangle is similar to triangle , which of the following ratios is equal according to the Law of Sines?
A
B
C
D
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Verified step by step guidance1
Recall that the Law of Sines states for any triangle ABC: \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\), where \(a\), \(b\), and \(c\) are the sides opposite angles \(A\), \(B\), and \(C\) respectively.
Since triangles \(\triangle qrs\) and \(\triangle xyz\) are similar, their corresponding angles are equal: \(Q\) corresponds to \(X\), \(R\) corresponds to \(Y\), and \(S\) corresponds to \(Z\).
Identify the sides opposite each angle in both triangles: side \(q\) is opposite angle \(Q\), side \(r\) opposite \(R\), side \(s\) opposite \(S\); similarly, side \(x\) opposite \(X\), \(y\) opposite \(Y\), and \(z\) opposite \(Z\).
Apply the Law of Sines to each triangle separately, then set the ratios of corresponding sides and sines of their opposite angles equal due to similarity: \(\frac{q}{\sin(Q)} = \frac{x}{\sin(X)}\), \(\frac{r}{\sin(R)} = \frac{y}{\sin(Y)}\), and \(\frac{s}{\sin(S)} = \frac{z}{\sin(Z)}\).
From these equalities, select the ratio that matches the given correct answer: \(\frac{q}{\sin(Q)} = \frac{x}{\sin(X)}\).
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