1.5: Visualisation
I found a (double) sheet from an old newspaper, with pages \(14\) and \(27\) next to each other. How many pages were there in the original newspaper?
A square \(ABCD\) of side \(2\) sits on top of a square \(PQRS\) of side \(1\), with vertex \(A\) at the centre \(O\) of the small square, side \(AB\) cutting the side \(PQ\) at the point \(X\), and \( \angle AXQ = \theta\).
(a) Calculate the area of the overlapping region.
(b) Replace the two squares in part (a) with two equilateral triangles. Can you find the area of overlap in that case? What if we replace the squares (i.e. regular \(4\)-gons) in part (a) with regular \(2n\)-gons?
The equilateral triangle \(\triangle ABC\) has sides of length \(1\) \(\text{cm}\). \(D\) and \(E\) are points on the sides \(AB\) and \(AC\) respectively, such that folding \(\triangle ADE\) along \(DE\) folds the point \(A\) onto \(A^{\prime}\) which lies outside \(\triangle ABC\).
What is the total perimeter of the region formed by the three single layered parts of the folded triangle (i.e. excluding the quadrilateral with a folded layer on top)?
The \(3\) by \(1\) rectangle \(ADEH\) consists of three adjacent unit squares: \(ABGH\), \(BCFG\), \(CDEF\) left to right, with \(A\) in the top left corner. Prove that
\(\angle DAE + \angle DBE = \angle DCE\)
(a) Joining the midpoints of the edges of an equilateral triangle \(ABC\) cuts the triangle into four identical smaller equilateral triangles. Removing one of the three outer small triangles (say \(AMN\), with \(M\) on \(AC\)) leaves three-quarters of the original shape in the form of an isosceles trapezium \(MNBC\). Show how to cut this isosceles trapezium into four congruent pieces.
(b) Joining the midpoints of opposite sides of a square cuts the square into four congruent smaller squares. If we remove one of these squares, we are left with three-quarters of the original square in the form of an L-shape. Show how to cut this L-shape into four congruent pieces.
The shaded region in Figure \(\PageIndex{1}\), shaped like a large comma, is bounded by three semicircles – two of radius \(1\) and one of radius \(2\).
Cut each region (the shaded region and the unshaded one) into two ‘halves’, so that all four parts are congruent (i.e. of identical size and shape, but with possibly different orientations).
Figure \(\PageIndex{1}\): Yin and Yang
In Problem 31 your first thought may have been that this is impossible. However, since the wording indicated that you are expected to succeed, it was clear that you must be missing something – so you tried again. The problem then tests both flexibility of thinking, and powers of visualisation.