Right-triangle sections in space

Finding the horizontal projection

A sloping line in a solid usually needs a triangle inside the solid. First identify the vertical height and the horizontal projection of the sloping line. Draw that right-triangle section separately.

For a vertical pole \(AT\) at a corner of a horizontal rectangle \(ABCD\), the line of sight from \(C\) is \(CT\). Its horizontal projection is the diagonal \(CA\), not a side of the rectangle.

\(AC^2=AB^2+BC^2\). Then, in the perpendicular section \(CAT\), \(\tan\angle TCA=\dfrac{AT}{AC}\).

A perpendicular height is not a sloping edge. In a pyramid, the section through a corner and the section through a side’s midpoint have different base lengths. Check the named angle before calculating.