1. In an equilateral Δ ABC, the side BC is trisected at D. Prove that 9AD2 = 7AB2
2. P and Q are points on sides AB and AC respectively, of ΔABC. If AP = 3 cm,PB = 6 cm, AQ = 5 cm and QC = 10 cm, show that BC = 3 PQ.
3. The image of a tree on the film of a camera is of length 35 mm, the distance from the lens to the film is 42 mm and the distance from the lens to the tree is 6 m. How tall is the portion of the tree being photographed?
4. . Prove that in any triangle the sum of the squares of any two sides is equal to twice the square of half of the third side together with twice the square of the median, which bisects the third side.
5. If a straight line is drawn parallel to one side of a triangle intersecting the othertwo sides, then it divides the two sides in the same ratio.
6. If ABC is an obtuse angled triangle, obtuse angled at B and if AD ^ CB Prove that AC2=AB2+ BC2+2 BC x BD
7. If a straight line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.
8. ABCD is a quadrilateral with AB =AD. If AE and AF are internal bisectors of ΔABC, D and E are points on AB and AC respectively such that AD/ DB = AEC/EC and ΔABC is isosceles.
9. In a ΔABC, points D, E and F are taken on the sides AB, BC and CA respectively such that DE IIAC and FE II AB.
10. . Prove that three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians of the triangle.
11. If a perpendicular is drawn from the vertex of a right angled triangle to its hypotenuse, then the triangles on each side of the perpendicular are similar to the whole triangle.
12. A man sees the top of a tower in a mirror which is at a distance of 87.6 m from the tower. The mirror is on the ground, facing upward. The man is 0.4 m away from the mirror, and the distance of his eye level from the ground is 1.5 m. How tall is the tower? (The foot of man, the mirror and the foot of the tower lie along a straight line).
13. In a right Δ ABC, right angled at C, P and Q are points of the sides CA and CB respectively, which divide these sides in the ratio 2: 1. Prove that
(I) 9AQ2= 9AC2+4BC2 (II) 9 BP2= 9 BC2+ 4AC2 (III) 9 (AQ2+BP2) = 13AB2
14. ABC is a triangle. PQ is the line segment intersecting AB in P and AC in Q such that PQ parallel to BC and divides Δ ABC into two parts equal in area. Find BP: AB.
15. P and Q are the mid points on the sides CA and CB respectively of triangle ABC right angled at C. Prove that4(AQ2+BP2) = 5 AB2
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