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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Key Concepts

Triangle Inequality Theorem

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This principle is crucial for determining whether a triangle can be formed with given lengths. In the context of the question, if the length L is too long relative to the other sides, it may violate this theorem, resulting in no triangle being formed.

Coordinate Geometry

Coordinate geometry involves the study of geometric figures using a coordinate system, typically the Cartesian plane. In this question, understanding how to represent points and line segments in relation to the x-axis is essential. The position of the point and the length L will determine the vertices of the triangle and whether they can form a valid shape.

Conditions for Triangle Formation

For a triangle to be formed, certain conditions must be met, including the lengths of the sides and their arrangement. Specifically, if the length L is equal to or greater than the distance from the point to the x-axis, it may not be possible to create a triangle. Recognizing these conditions helps in identifying scenarios where no triangle can be drawn.