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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblema 3c
Capitolo 8, Problema 3c

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?
c. no triangle
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Guida verificata passo dopo passo
1
Identify the given point's coordinates and the angle it makes with the positive x-axis, if provided, or determine the vertical distance from the point to the x-axis.
Recall that the line segment of length \(L\) is drawn from the point to the positive x-axis, forming a triangle with the x-axis and the segment from the origin to the foot of the perpendicular.
Understand that a triangle can be formed only if the length \(L\) is greater than the shortest distance from the point to the x-axis; if \(L\) is less than this distance, no triangle can be formed.
Express the shortest distance from the point to the x-axis mathematically, which is the absolute value of the y-coordinate of the point, say \(d = |y|\).
Conclude that for no triangle to be formed, the length \(L\) must satisfy \(L < d\), meaning the segment is too short to reach the x-axis and form a triangle.

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Triangle Inequality Theorem

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. This principle helps determine whether a triangle can be formed given certain side lengths, ensuring the segments can connect to form a closed shape.
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Solving Right Triangles with the Pythagorean Theorem

Distance from a Point to the x-axis

The distance from a point to the x-axis is the absolute value of the point's y-coordinate. This distance is crucial when drawing a segment from the point to the x-axis, as it sets a minimum length for the segment and influences the possible lengths L that can form a triangle.
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Convert Points from Rectangular to Polar

Conditions for No Triangle Formation

No triangle is formed when the segment length L violates the triangle inequality, such as being too short or too long relative to other sides. Understanding these conditions helps identify values of L for which a triangle cannot exist, often involving equality or impossible side length combinations.
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Evaluating Sums and Differences Given Conditions
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