This problem is based on the concept of similar triangles, which can be used to calculate the height of the football goalpost.

Step 1: Understanding the geometry

Violet and the top of the goalpost are forming two similar triangles with the mirror as the point where the light rays reflect. Here’s what we know:

1. The distance from the goalpost to the mirror is 6.65 meters.
2. The distance from Violet to the mirror is 4.25 meters.
3. Violet's eye height from the ground is 1.75 meters.
4. We need to find the height of the goalpost, which we will call $h$.

Since the triangles are similar, the ratio of corresponding sides is equal: $\frac{h}{6.65} = \frac{1.75}{4.25}$

Step 2: Set up the equation

Using the proportional relationship of similar triangles, we can set up the equation as follows: $\frac{h}{6.65} = \frac{1.75}{4.25}$

Step 3: Solve for $h$

Multiply both sides of the equation by 6.65 to isolate $h$: $h = \frac{1.75}{4.25} \times 6.65$

Now, let’s calculate the value of $h$.The height of the goalpost is approximately 2.74 meters, rounded to the nearest hundredth.

Would you like more details, or have any questions on the process?

Here are 5 related questions you could explore:

1. How are similar triangles used in real-life measurements like this?
2. What is the importance of mirror reflections in geometric problems?
3. How would the calculation change if the distances were different?
4. How can we verify the accuracy of the height measurement using another method?
5. Could this technique work if Violet were at a different height?

Tip: Always ensure the angles in similar triangle problems are consistent to apply proportional reasoning correctly.