The diagram shows a large semicircle with diameter `AB` and two smaller semicircles with diameters `AC` and `BC`, respectively, where `C` is a point on the diameter `AB`. The point `M` is the centre of the semicircle with diameter `AC`.
The line perpendicular to `AB` through `C` meets the largest semicircle at the point `D`. The points `S` and `T` are the intersections of the lines `AD` and `BD` with the smaller semicircles. The point `X` is the intersection of the lines `CD` and `ST`.
Explain why `CTDS` is a rectangle. (3 marks)
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`/_SDT = 90°\ text{(angle in semi-circle)}`
`/_ ASC = 90°\ \ text{(angle in semi-circle)}`
`=> /_ CSD = 180-90=90°\ \ text{(} /_ ASD\ text{is a straight line)}`
`text(Similarly,)`
`/_CTB = /_CTD=90°`
`/_SCT = 90°\ \ text{(angle sum of quadrilateral}\ CTDS text{)}`
`text(S)text(ince all angles are right angles,)`
`CTDS\ text(is a rectangle)`