Q.1. Let AOB be a given angle less than and let P be an interior point of the angular region determined by . Show, with proof, how to construct, using only ruler and compass, a line segment CD passing through P such that C lies on the way OA and D lies on the ray OB, and CP:PD=1:2.
Analysis: Draw a line parallel to OB through C. Let it intersect OP (extended) at K. and are similar which implies
Construction: Choose K on extended OP such that OP : PK=2:1. Draw a line parallel to OB through K. It will intersect OA at C (required). Then CP will cut OB at D (required).
Q.5. Let be a circle with a chord AB which is not a diameter. be a circle on one side of AB such that it is tangent to AB at C and internally tangent to at D. Likewise, let be a circle on the other side of AB such that it is tangent to AB at E and internally tangent to at F. Suppose the line DC intersects at and the line FE intersects at . Prove that XY is a diameter of
Label the centres as P and Q, as shown in the figure. Let O be the centre of the main circle.
gives Now, so we get Thus XO is nothing but the perpendicular bisector of AB. Similarly YO is also the perpendicular bisector of AB. Hence X, O, Y are collinear. Hence done.