In a regular quadrangular prism through the diagonal of the base, a section is drawn parallel to the diagonal

| September 25, 2021 | education

In a regular quadrangular prism through the diagonal of the base, a section is drawn parallel to the diagonal of the prism. Find the cross-sectional area if the side of the base of the prism is 2cm and its height is 4.

Let us construct the required section BDM and draw a segment OM in the triangle BDM.

By condition, the section plane is parallel to the AC1 diagonal, and therefore parallel to the OM segment. Since point O divides the diagonals in half, OA = OC, and therefore OM is the middle line of the ACC1 triangle, then CM = CC1 / 2 = 4/2 = 2 cm.

Let us define the diagonals AC and BD at the base of the prism.

AC ^ 2 = BD ^ 2 = AD ^ 2 + CD ^ 2 = 4 + 4 = 8.

AC = BD = 2 * √2 cm.

Then OS = AC / 2 = 2 * √2 / 2 = √2 cm.

In a right-angled triangle OCM, according to the Pythagorean theorem, OM ^ 2 = OC ^ 2 + CM ^ 2 = 2 * 4 = 6.

ОМ = √6 cm.

Determine the cross-sectional area. Ssection = ВD * ОМ / 2 = 2 * √2 * √6 / 2 = √12 = 2 * √3 cm2.

Answer: The cross-sectional area is 2 * √3 cm2.



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