The length of the transverse axis is 11, and the length of the red line segment is 19. How long is the blue line segment?

1. Break Down the Problem

To find the length of the blue line segment, we need to understand the relationship between the red line segment, the transverse axis, and the blue line segment. The transverse axis typically refers to the main "horizontal" distance in an ellipse or hyperbola.

2. Relevant Concepts

If we interpret this problem with respect to a hyperbola, the transverse axis ($2a$) represents the distance across the hyperbola along its major axis, where $a$ is half the length of the transverse axis. The red line segment could represent the distance from the center to one of the vertices, and the blue line segment may refer to the distance from the center to the other vertex.

3. Analysis and Detail

• The length of the transverse axis is given as 11, therefore $a = \frac{11}{2} = 5.5$.
• The red line segment, which we can assume to represent the distance from the center to one of the vertices, is 19.

If these segments are part of a hyperbola, we can infer that the relationship can be defined as:

For hyperbolas: $c^2 = a^2 + b^2$ Where:

• $c$ is the distance from the center to the foci (which could potentially match the length of the red line segment).
• $b$ represents the distance related to the other dimensions of the hyperbola.

However, given the data we have:

• $c = 19$
• $a = 5.5$

Using the above relation: $19^2 = (5.5)^2 + b^2$ $361 = 30.25 + b^2$ $b^2 = 361 - 30.25 = 330.75$ $b = \sqrt{330.75} \approx 18.2$

4. Verify and Summarize

We calculated $b$ and found it to be approximately 18.2. This value represents the blue line segment length if we are interpreting it in the hyperbolic context correctly.

Final Answer

The length of the blue line segment is approximately $18.2$.