Question

What is the shape of the cross-section formed when a plane containing line AC and line EH intersects this cube? What is the area of the cross-section? In two or more complete sentences, explain how you were able to determine the shape and area of the cross-section. Be sure to include the properties of quadrilaterals and the mathematical computations necessary to support your answers

Answer 1

Given a cube of sides 4 units with vertices: A, B, C, D, E, F, G, H such that the top square is ABCD and the bottom square is EFGH with AC, BD the diagonals of the top square and EG, FH the diagonals of EFGH.

The shape of the cross-section formed when a plane containing line AC and line EH intersects the cube is the rectangle ACEH.

We obtain the value of line AC which is equal to line EH using the pythagoras rule which states that $c^2=a^2+b^2$, where c is the length of line AC, a is the length of line AB = 4 and b is the length of line BC = 4.

Thus, 
$c^2=4^2+4^2=16+16=32 \\ c= \sqrt{32} =4 \sqrt{2}$

Therefore, the shape of the cross-section formed when a plane containing line AC and line EH intersects the cube is the rectangle ACEH with length of $4 \sqrt{2}$ units and width of 4 units.

Area of a rectangle is given by
Area = length x width
$=4 \sqrt{2}\times4=16 \sqrt{2}$ square units.