91Ó°ÊÓ

Problem 142 Show how to implement the basic ruler and compasses constructions: (i) to construct the midpoint \(M\) of a given line segment \(\underline{A B}\); (ii) to bisect a given angle \(\angle A B C\); (iii) to drop a perpendicular from \(P\) to a line \(A B\) (that is, to locate \(X\) on the line \(A B\), so that the two angles that \(P X\) makes with the line \(A B\) on either side of \(P X\) are equal). Prove that your constructions do what you claim.

Let \(O\) be the circumcentre of \(\triangle A B C\). Prove that $$\angle A O B=2 \cdot \angle A C B$$

Let \(A B C D\) be a quadrilateral inscribed in a circle (such a quadrilateral is said to be cyclic, and the four vertices are said to be concyclic that is, they lie together on the same circle). Prove that opposite angles (e.g. \(\angle B\) and \(\angle D)\) must add to a straight angle. (Two angles which add to a straight angle are said to be supplementary.)

The point \(P\) lies outside a circle. Two secants from \(P\) meet the circle at \(A, B\) and at \(C, D\) respectively. Prove in two different ways that $$\underline{P A} \times \underline{P B}=\underline{P C} \times \underline{P D}$$

(a) Find the exact area (in terms of \(\pi\) ) (i) of a semicircle of radius \(r\); (ii) of a quarter circle of radius \(r\) (iii) of a sector of a circle of radius \(r\) that subtends an angle \(\theta\) radians at the centre. (b) Find the area of a sector of a circle of radius \(1,\) whose total perimeter (including the two radii) is exactly half that of the circle itself.

The only possible path along the edges of a 2 D-cube uses each vertex once and returns to the start after visiting all four vertices. (a)(i) Draw a path along the edges of a 3D-cube that visits each vertex exactly once and returns to the start. (ii) Look at the sequence of coordinate triples as you follow your path. What do you notice? (b)(i) Draw a path along the edges of a \(4 \mathrm{D}\) -cube that visits each vertex exactly once and returns to the start. (ii) Look at the sequence of coordinate 4 -tuples as you follow your path. What do you notice?