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. In a rectangular prism, when we consider the cross - section through \(H\), \(I\), \(N\), and \(O\), we know that in a rectangular prism, opposite sides are parallel. \(HI\parallel ON\) (since \(HI\) and \(ON\) are both parallel to the edges of the top and bottom faces of the prism) and \(HO\parallel IN\) (by the properties of a rectangular prism). So, the cross - section is a parallelogram. Also, since the angles in a rectangular prism are right angles, \(HI\perp IN\), so the cross - section is a rectangle.

Step2: Determine the dimensions of the cross - section

The length of \(HI\) is equal to the length of \(ON\). In the base of the prism, \(ON = 3\space cm\). The length of \(IN\) (or \(HO\)) can be found using the Pythagorean theorem in the right - triangle formed by the height of the prism (\(4\space cm\)) and the diagonal of the square base. The base is a square with side \(a = 3\space cm\). The diagonal of the base \(d=\sqrt{3^{2}+3^{2}}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\space cm\). But if we consider the cross - section \(HINO\), \(HI = 3\space cm\) (same as the side of the base) and \(IN=\sqrt{3^{2}+4^{2}}\) (using the right - triangle with legs \(3\space cm\) (side of the base) and \(4\space cm\) (height of the prism)). By the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 3\) and \(b = 4\). So \(IN=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5\space cm\)

Answer:

The cross - section passing through vertices \(H\), \(I\), \(N\), and \(O\) is a rectangle with dimensions \(3\space cm\times5\space cm\)