To determine the measure of $GA$, let's analyze the provided geometry.

Analysis:

1. Observations from the diagram:
   • $G$, $A$, $B$, $D$, $E$, and $F$ are points in a complex figure involving multiple triangles.
   • $GA$ is perpendicular to $AB$ and forms a right angle at $G$.
   • $AB = 3$, and $BG = 4$ (from the diagram).
   • $AG$ is the hypotenuse of the right triangle $\triangle ABG$.

Step-by-step Solution:

We use Pythagoras' Theorem, which states: $\text{In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.}$ For $\triangle ABG$: $GA^2 = AB^2 + BG^2$

Substituting the values: $GA^2 = 3^2 + 4^2$ $GA^2 = 9 + 16 = 25$

Taking the square root: $GA = \sqrt{25} = 5$

Final Answer:

$GA = 5$

Formula/Property Used:

The calculation used Pythagoras' Theorem for right triangles. Specifically: $\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2$

Would you like me to explain further or explore additional aspects of the geometry in this diagram?

Related Questions:

1. How can we verify the lengths of other segments, such as $EF$ or $DE$?
2. What trigonometric relationships exist in the angles provided (e.g., $\angle F$, $\angle E$)?
3. Can the area of $\triangle ABG$ or the entire shape be calculated?
4. How do the angles $\angle F$ and $\angle C$ relate to the geometry of the diagram?
5. Could the slopes of the lines in the diagram be used to find segment relationships?

Tip:

Always identify right triangles within complex diagrams; they often simplify calculations using Pythagoras' Theorem or basic trigonometric relationships.