CBSE TEST PAPER MATHEMATICS (Class-10) SIMILAR TRIANGLE

1. In Δ PQR, given that S is a point on PQ such that ST II QR and PS/SQ=3/5 If PR = 5.6 cm, then find PT.

2. In Δ ABC, AE is the external bisector of <A, meeting BC produced at E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, then find CE.

3. 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.

4. 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?

5. D is the midpoint of the side BC of Δ ABC. If P and Q are points on AB and on AC such that DP bisects <BDA and DQ bisects <ADC, then prove that PQ II BC.

6. If a straight line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.

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 <BAC and <DAC respectively, then prove that EF II BD. In a Δ ABC, D and E are points on AB and AC respectively such that AD/ DB = AEC/EC and <ADE = <DEA. Prove that Δ ABC is isosceles.

9. In a Δ ABC, points D, E and F are taken on the sides AB, BC and CA respectively such that DE IIAC and FE II AB.

10. The internal bisector of <A of ΔABC meets BC at D and the external bisector of <A meets BC produced at E. Prove that BD/ BE = CD/CE

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 that 4(AQ² +BP²) = 5 AB²

16. In an equilateral Δ ABC, the side BC is trisected at D. Prove that 9AD2 = 7AB2

17. 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.

18. If ABC is an obtuse angled triangle, obtuse angled at B and if AD^CB Prove that 

1.    AC² =AB² + BC²+2 BC x BD

19. 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. 

 [To prove AB² + AC² = 2AD² + 2(1/2BC)²]

20. ABC is a right triangle right-angled at C and AC= √3 BC. Prove that <ABC= 60^o