To determine the coordinates of the vertices of the rectangle, let's break down the given information:

1. Length of the rectangle (longer side): 5 units
2. Breadth of the rectangle (shorter side): 3 units
3. One vertex is at the origin: $(0, 0)$
4. The longer side lies on the x-axis.
5. One of the vertices lies in the third quadrant.

Given that the rectangle lies along the x-axis with one vertex at the origin and has a vertex in the third quadrant, the rectangle's orientation will be such that one side is parallel to the x-axis, and one side is parallel to the y-axis.

Vertices:

• Vertex A: Since it is at the origin, the coordinates are $(0, 0)$.
• Vertex B: Since the longer side lies on the x-axis, and its length is 5 units, the coordinates are $(5, 0)$.
• Vertex C: Moving 3 units downward (negative y-direction) from Vertex B because a vertex lies in the third quadrant, the coordinates are $(5, -3)$.
• Vertex D: Moving 3 units downward from Vertex A along the negative y-axis, the coordinates are $(0, -3)$.

Therefore, the coordinates of the vertices of the rectangle are:

• $A = (0, 0)$
• $B = (5, 0)$
• $C = (5, -3)$
• $D = (0, -3)$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the coordinates change if the rectangle were rotated 90 degrees?
2. What are the midpoints of the sides of this rectangle?
3. How would you calculate the area of this rectangle using its vertex coordinates?
4. What would the coordinates of the vertices be if the rectangle were translated 2 units to the left and 4 units up?
5. How can you determine if a point lies inside this rectangle?

Tip: Always sketch a diagram when working with coordinate geometry problems to visualize the relationships between points.