155. Three angle sum


In the diagram, one side of a rectangle is divided into three equal parts.
Determine the sum of angles x, y and z.


154. Line segment sweep


Points A(0,4), B(–2,0), C(6,0) are the vertices of △ABC.  Point P is on side AC.  A straight line extends from the origin O(0,0) through point P to point Q so that the area of △ABO=the area of △QAC.  If point P travels from point A to point C, determine the area of the shape that line segment PQ traces.

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153. Triangle + semicircles, stir


In the diagram, the two legs of right triangle ABC form the diameters of two semicircles.

⒜  Determine the total area of the shaded regions (coloured regions).
* Prove (as shown in the diagram) that one point of intersection of the two semicircles lies on the hypotenuse.