The problem involves finding the coordinates of points $A$, $B$, and $C$ on a parabola that represents the arch of a bridge. Let's solve it step by step.

Step 1: Understanding the diagram

• The distance between points $A$ and $C$ is given as 8 meters, which means the length of the arch between these two points is $AC = 8$ meters.
• The total length of the bridge is 14 meters.
• The height of the arch at its highest point, $B$, is 5 meters.

Step 2: Coordinate setup

• The problem states that the arch is symmetric and parabolic, and it provides a coordinate system where the origin $O(0, 0)$ is at point $A$.
• Point $A$ is on the x-axis, point $B$ is the highest point on the arch (the vertex), and point $C$ is symmetrically located on the other side.

From the figure, we can interpret:

• Point $A$ is at the origin: $A(0, 0)$.
• Point $C$, located symmetrically, is 8 meters from point $A$ along the x-axis: $C(8, 0)$.
• Point $B$, at the vertex of the parabola, is at the midpoint of $AC$, which is 4 meters from $A$, and its height is 5 meters. Therefore, the coordinates of point $B$ are: $B(4, 5)$.

Step 3: Coordinates of points

• Point $A = (0, 0)$
• Point $B = (4, 5)$
• Point $C = (8, 0)$

Thus, the coordinates of points $A$, $B$, and $C$ are:

• $A(0, 0)$
• $B(4, 5)$
• $C(8, 0)$

Would you like more details on finding the equation of the parabola, or do you have any questions?

Expanding Questions:

1. How do you derive the equation of a parabola from three points?
2. How can we find the equation of the parabola given these points?
3. How does symmetry help in simplifying parabolic problems?
4. What are the other forms of parabolas based on their orientation?
5. How does the vertex form of a parabola relate to this scenario?

Tip:

The equation of a parabola can be written in vertex form as $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.