Question

the rectangular prism below has two congruent square faces and four congruent rectangular faces.

the line segment formed by connecting vertices e and b is 18 units long, the line segment formed by connecting vertices g and e is 13 units long, and the line segment formed by connecting vertices h and a is 18 units long.

if the rectangular prism is sliced by a plane that passes through vertices h, c, and a, which of the following best describes the resulting cross - section of the prism?

a. a triangle with a height of 18 units and a base of 13 units

b. a rectangle with a length of 18 units and a width of 13 units

c. a square with side lengths of 18 units

d. a triangle with side lengths of 18 units, 13 units, and 18 units

Explanation:

Step1: Analyze the cross - section

When a plane slices through three non - collinear vertices \(H\), \(C\), and \(A\) of a rectangular prism, the resulting cross - section is a triangle.

Step2: Determine the side lengths

We know that \(EB = 18\) units, \(GE=13\) units, \(HA = 18\) units. In a rectangular prism, \(HA\) and \(EB\) are equal in length (because of the congruent square and rectangular faces properties). Also, \(GE\) is another side. The side lengths of the triangle formed by connecting \(H\), \(C\), and \(A\) are \(HA = 18\) units, \(HC=EB = 18\) units (since \(HC\parallel EB\) and \(HC = EB\) in the prism structure), and \(CA=GE = 13\) units (since \(CA\parallel GE\) and \(CA = GE\) in the prism structure)

Answer:

D. a triangle with side lengths of 18 units, 13 units, and 18 units