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Chapter 44: Problem 812

Show that the lateral edges of a regular pyramid are congruent.

Short Answer

Expert verified
To show that the lateral edges of a regular pyramid are congruent, consider a triangular face consisting of the vertex V and two adjacent vertices A and B of the base. Triangle VAB is an isosceles triangle with VA and VB as congruent sides. Since the base is a regular polygon, each vertex is equidistant from the centroid C, so CA = CB. As V is the apex of the pyramid and distances VC and VB are equal, we conclude that VA = VB. Applying the same reasoning for all adjacent triangles sharing the apex V, we can ultimately prove that all lateral edges are congruent.

Step by step solution

01

1. Understanding a Regular Pyramid

A regular pyramid is a solid figure having a regular polygon as its base and triangular lateral faces that all meet at a common vertex. The key property of a regular pyramid is that its base is a regular polygon, and all its lateral edges have the same length.
02

2. Defining Given Information

We are given a regular pyramid with a base of a regular polygon. Let the base be a regular polygon with n sides, and let the common vertex of the pyramid be V.
03

3. Identifying the Lateral Edges

The lateral edges are the edges connecting the vertices of the pyramid's base to the common vertex V. There are n lateral edges in the pyramid, corresponding to the n vertices of its base.
04

4. Finding the Length of a Lateral Edge

To find the length of a lateral edge, let's consider a triangular face of the pyramid, consisting of vertex V and two adjacent vertices A and B of the base. The triangle VAB is an isosceles triangle, with VA and VB as its congruent sides (the lateral edges).
05

5. Prove Lateral Edges Are Congruent

Since the base of the pyramid is a regular polygon, the distance of each vertex to the centroid of the base is the same. Let's denote the centroid as C. Now, in the triangle VAC, we have VA = VC + CA. Since the base is a regular polygon, the vertices are equidistant from the centroid C, so CA = CB. Moreover, since V is the apex of the pyramid and the distances VC and VB are equal (because it is a regular pyramid), we can conclude that VA = VB. We can apply the same reasoning for all adjacent triangles sharing the apex V and edges of the base polygon and ultimately prove that all lateral edges are congruent.

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Most popular questions from this chapter

Show that the locus of points equidistant from two given points is the plane perpendicular to the line segment joining them at their midpoint.

Show that if a plane intersects two parallel planes, then it intersects them in two parallel lines.

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