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
Problem 3b
Textbook Question
In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?
b. exactly one triangle
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Verified step by step guidance1
Identify the given point's coordinates and the angle(s) involved in forming the triangle with the positive x-axis and the segment of length \(L\).
Recall that the problem is about drawing a segment of length \(L\) from the point to the x-axis to form a triangle, so consider the geometric constraints: the segment must connect the point to some point on the positive x-axis.
Use the distance formula or trigonometric relationships to express the length \(L\) in terms of the coordinates of the point and the position on the x-axis where the segment meets it.
Analyze the conditions under which exactly one triangle can be formed. This typically happens when the segment length \(L\) equals the perpendicular distance from the point to the x-axis, or when the segment is tangent to a circle centered at the point with radius \(L\).
Set up an equation relating \(L\) to the coordinates and solve for the value(s) of \(L\) that yield exactly one solution, indicating exactly one triangle can be formed.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Triangle Formation Conditions
Understanding when a triangle can be formed involves analyzing the lengths of sides and their relationships. Specifically, the triangle inequality theorem states that the sum of any two sides must be greater than the third side. This concept helps determine if a given segment length L can form one, two, or no triangles with the given points.
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Evaluating Sums and Differences Given Conditions
Geometric Interpretation of Segment Lengths
Drawing a segment of length L from a point to the x-axis involves visualizing possible positions of the segment’s endpoint. The number of triangles formed depends on how many distinct points on the x-axis satisfy the length condition, which can be interpreted using circles or arcs centered at the given point with radius L.
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Adding Vectors Geometrically
Uniqueness of Triangle Formation
Exactly one triangle is formed when the segment length L corresponds to a unique intersection point on the x-axis. This typically occurs when the segment length equals the perpendicular distance from the point to the x-axis or when the segment is tangent to the circle representing all points at distance L, ensuring a single solution.
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Related Practice
Textbook Question
In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.a = 30, b = 20, A = 50°
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