Question

Let AOB be a given angle less than 180° and let P be an interior point of the angular region determined by ZAOB.

Show, with proof, how to construct, using only ruler and compasses, a line segment CD passing through P such that C lies on the ray OA and D lies on the ray OB, and CP: PD=1:2.​

Answer 1

$\huge\tt\bold\green{Answer :-}$

Join OP and extend to Q such that OP : PQ = 2 : 1 Draw a line parallel to OA through Q.

It cuts OB at point M (say) take point D on OB such that M is mid point of OD joint DQ and Produce it to meet OA at N. Then by mid-point theorem, Q is mid point of DN. so OQ is median of

Δ ODN

• As OP : PQ = 2 : 1,
• P is centroid => DP : PC = 2 : 1

Construction of OP : PQ = 2 : 1

• obtain mid-point L of OP
• with P as centre and radius PL, arc cuts OP produced at Q

Construction of QM || OA :

• With O as centre, draw an arc to cut OQ at S1 and OA at SQ
• With Q as centre and same radius, draw an arc to cut OQ at S3
• With S3 as centre and radius S1 S2, draw an arc to cut previous arc at S4 join QS4