1. InΔPQR, 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. InΔABC, 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, ofΔABC. 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 distancefrom the lens to the film is 42 mm andthe distancefrom 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 ofΔABC. 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 othertwo sides, then it divides thetwo sidesin the same ratio.
7. If a straight line divides anytwo sidesof 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 aΔABC, D and E are points on AB and AC respectively such that AD/ DB = AEC/EC and <ADE = <DEA. Prove thatΔABC is isosceles.
9. In aΔABC, 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 ofΔABC 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 maway fromthe mirror, andthe distanceof 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 rightΔABC, 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 dividesΔABC 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
16. In an equilateralΔABC, 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 ofthe 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 anytwo sidesis 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= 60o
In ΔABC, right angled at B, BD⊥AC
To prove : ΔADB, ΔBCD are similar to each other and also similar to triangle ABC.
(1) In ΔABC, ∠B = 90°
⇒ ∠BAC + ∠BCA = 90° ..... (1)
In ΔDBC, ∠D = 90°
⇒ ∠DBC + ∠BCD = 90° ..... (2)
Since, ∠BCD = ∠BCA
∴ From (1) and (2), we get –
∠BAC = ∠DBC
⇒ ∠BAD = ∠DBC ..... (3)
Now in ΔADB and ΔBDC, we have
∠BAD = ∠DBC and ∠ADB = ∠BDC = 90°
∴ ΔABD is similar to ΔBDC [AA– similarity]
(2) In ΔADB and ΔABC,
∠ABC = ∠ADB = 90°
∠A is common
∴ ΔADB and ΔABC are similar
(3) In ΔBDC and triangle ABC,
∠BDC = ∠ABC = 90°
∠C is common
∴ ΔABC and ΔBDC are similar