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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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 when determining the possible lengths of a line segment L that can form a triangle with the x-axis and another line segment. Understanding this theorem helps in identifying the conditions under which exactly one triangle can be formed.

Conditions for Triangle Formation

To form a triangle, certain conditions must be met regarding the lengths of the sides. Specifically, if one side is fixed (like the segment to the x-axis), the other two sides must be able to connect at a point without violating the triangle inequality. This concept is essential for determining the specific value of L that allows for the formation of exactly one triangle.

Geometric Interpretation of L

The length L can be interpreted geometrically as the height of a triangle formed with the x-axis. When L is equal to the distance from the point to the x-axis, it creates a right triangle. Understanding this geometric relationship is key to visualizing how varying L affects the number of triangles that can be formed, particularly in scenarios where only one triangle is possible.