Question

this is a rectangular prism.
use the prism to complete the sentence.
the cross section passing through vertices h, i, n, and o is a with dimensions

Explanation:

Step1: Analyze the shape of the cross - section

A cross - section passing through four vertices of a rectangular prism. Since \(HI\parallel ON\) (because \(HI\parallel KM\) and \(ON\parallel KM\) in a rectangular prism) and \(HO\parallel IN\) (by the properties of a rectangular prism), the cross - section is a parallelogram. Also, in a rectangular prism, adjacent edges are perpendicular. \(HI = 3\mathrm{cm}\), \(IN=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5\mathrm{cm}\) (using the Pythagorean theorem in right - triangle \(INM\) where \(NM = 3\mathrm{cm}\) and \(IM=4\mathrm{cm}\)). But if we consider the fact that \(HI\perp HO\) (because \(HI\perp\) the base \(OKMN\) and \(HO\) is on the base - adjacent face). So, the cross - section is a rectangle.

Step2: Determine the dimensions of the cross - section

The length of \(HI = 3\mathrm{cm}\). The length of \(HO\): In right - triangle \(HOL\) (where \(OL = 3\mathrm{cm}\) and \(LH=4\mathrm{cm}\)), using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), \(HO=\sqrt{3^{2}+4^{2}} = 5\mathrm{cm}\)

Answer:

The cross - section passing through vertices \(H\), \(I\), \(N\), and \(O\) is a rectangle with dimensions \(3\mathrm{cm}\) and \(5\mathrm{cm}\)