A three dimensional solid is a triangular prism that has a total of 9 edges. Based on the properties of prisms, determine how many vertices this triangular prism has.

Introduction / Context:
This geometry question is about basic properties of three dimensional solids, specifically a triangular prism. A triangular prism has two congruent triangular faces connected by three rectangular faces. Understanding the relationship between faces, edges, and vertices in common solids is important in spatial reasoning and is often tested in aptitude and school level exams.

Given Data / Assumptions:

• The solid in question is a triangular prism.
• The prism has 9 edges.
• We are asked to find the total number of vertices.
• The prism is assumed to be a simple, right or oblique, triangular prism without any truncation.

Concept / Approach:
A triangular prism can be visualised as a triangle extruded along a straight line. It has two triangular bases and three rectangular lateral faces. Each triangular base has 3 vertices and 3 edges. The corresponding vertices of the two triangles are connected by 3 additional edges along the length of the prism. Counting these features carefully allows us to determine the total number of vertices, edges, and faces.

Step-by-Step Solution:
Step 1: Consider one triangular base. A triangle has 3 vertices and 3 edges.Step 2: Since the prism has two congruent triangular bases, the total number of vertices from both triangles is 3 + 3 = 6.Step 3: The edges on the two bases contribute 3 edges on the top triangle and 3 edges on the bottom triangle, giving 6 base edges.Step 4: The remaining edges of the prism connect corresponding vertices of the two triangles. There are 3 such connecting edges.Step 5: Therefore, the total number of edges is 3 + 3 + 3 = 9, which matches the given information.Step 6: We have already counted the number of vertices as 3 on each triangular base, so the total number of vertices is 6.

Verification / Alternative check:
We can also recall the general structure of a triangular prism: it always has 5 faces (2 triangular and 3 rectangular), 9 edges, and 6 vertices. This is a standard fact from solid geometry. Since the problem statement confirms that the solid has 9 edges, we can safely apply this known property to conclude that it has 6 vertices.

Why Other Options Are Wrong:
Eight vertices would be appropriate for a rectangular prism (a cuboid), not a triangular prism. Twelve vertices correspond to more complex polyhedra with additional vertices, not to a simple triangular prism. Ten and four vertices do not match any standard prism built on a triangle. Only six vertices are consistent with a prism whose base is a triangle and which has exactly 9 edges.

Common Pitfalls:
Some learners confuse triangular prisms with rectangular prisms or cubes and incorrectly assume 8 vertices. Others miscount by forgetting that each vertex on one triangular base has exactly one corresponding vertex on the other base. Drawing a quick sketch of a triangular prism and marking all vertices and edges is an effective way to prevent mistakes and confirm the counts.

Final Answer:
The triangular prism with 9 edges has 6 vertices.