In triangle PQR, the line XY is drawn parallel to QR, meeting PQ at X and PR at Y. If PX = 4 cm, XQ = 6 cm and PY = 5 cm, find the length of YR.

In triangle ABC, D is a point on AB and E is a point on AC such that DE || BC. If AD = 5 cm, AB = 15 cm and BC = 18 cm, find the length of DE.

In triangle LMN, PM = 6 cm, MN = 8 cm, LN = 10 cm. Points A and B are on LM and LN respectively such that LA = 3 cm and LB = 5 cm. Is AB parallel to MN? Give a reason.

Triangles ABC and PQR are equiangular. If AB = 6 cm, BC = 8 cm, AC = 10 cm and PQ = 9 cm, find the lengths of QR and PR.

In triangle ABC, AB^C=90∘. D is a point on AC such that BD⊥AC. Prove that triangle ABD is equiangular to triangle BCD.

A vertical pole 6 m tall casts a shadow 4 m long. At the same time, a nearby tower casts a shadow 28 m long. Find the height of the tower.

In the figure, AB || DE. AC = 4 cm, CD = 6 cm, BC = 5 cm. Find the length of CE.

State the two conditions for two polygons to be similar.

In trapezium ABCD, AB || DC. The diagonals AC and BD intersect at O. Prove that triangle AOB is similar to triangle COD.

Using the result from question 9, if AO = 3 cm, OC = 6 cm and the area of triangle AOB is 12 cm², find the area of triangle COD.

In triangle ABC, P is a point on AB and Q is a point on AC. If AP^Q=AC^B, prove that triangle APQ is similar to triangle ACB.

In triangle PQR, S is a point on QR such that PS^R=QP^R. Prove that PR2=QR⋅RS.

A line drawn through the vertex A of a parallelogram ABCD meets the diagonal BD at X, the side BC at Y and DC produced at Z. Prove that AX is the mean proportional between XY and XZ.

In triangle ABC, AD^C=BA^C. If AC = 12 cm and DC = 8 cm, find the length of BC.

Prove that if a straight line divides two sides of a triangle proportionally, then that line is parallel to the remaining side (Converse of Proportionality Theorem).