Ch8_ Similar Triangles Main

ASSIGNMENT CLASS X SIMILAR TRIANGLES Q1. In ∆ABC, DE || BC (a) IF AD /DB = 2/3 and AC = 18cm, find AE. (b) IF AD = x, DB = x – 2 , AE = x + 2, EC = x –1, find x. (c) If AD = 8cm, AB = 12cm, AE = 12cm, find CE. Q2. In the given figure, AB || DC. If EA = 3x – 19, EB = x – 4, EC = x – 3 and ED = 4, find x. D

C E B

A

Q3. In the following figure, DE || BC and AD:DB = 5:4, find ar(∆DFE)/ar(∆CFB) A D E F

B

C

Q4. In DABC, D is a point on AB, E is a point on AC and DE || BC. If AD : DB = 5 : 4, find : (i) DE, if BC = 27 cm (ii) ar (DADE) : ar (DABC)

(iii) ar (DADE) : ar (trapezium DBCE)

A D

E C

B

Q5. In the given figure, PA, QB and RC are perpendicular to AC. Show that

1 1 1   x y z

P x

R

Q

y

z A

a

b

B

C

Q6. In the given figure, DEFG is a square and BAC = 90°. Prove that : (i) DAGF ~ DDBG

(ii) DAGF ~ DEFC

(iii) DDBG ~ DEFC

(iv) DE2 = BD × EC

A G

B

D

F

E

C

Q7. Two triangles BAC and BDC, right angled at A and D respectively, are drawn on same base BC and on same side of BC. If AC and DB intersects at P, prove that AP X PC = DP X BP. Q8. Two poles of height a meters and b meters are p meters apart. Prove that height of the point of ab intersection of line joining top of each pole to foot of the opposite pole is meters. a+b Q9. Prove that the areas of two similar triangles are in the ratio of the squares of their corresponding: (a) altitudes (b) medians (c) angle bisectors

Q10. In the given figure, find x in terms of a, b and c. A E

a x

53°

B

b

53°

C

c

D

Q11. The perimeters of two similar triangles are 25cm and 15cm respectively. If one side of first triangle is 8cm, what is corresponding side of the other triangle?

6c m

Q12. ABC is an isosceles triangle right angled at B. Similar triangles ACD and ABE are constructed on sides AC and AB. Find the ratio between the areas of ∆ABE and ∆ACD. Q13. In a right angled triangle ABC,  A = 90° and AD  BC. Prove that AD2 = BD × CD. Q14. In a right angle triangle ABC, right angled at C, P and Q are the points of the sides CA and CB respectively which divides these sides in the ratio 1:2, prove that 9(AQ2 + BP2) = 12AB2 Q15. In right ∆ABC, right angled at C, a point D is taken on AB such that CD is perpendicular to AB. 1 1 1 Prove that + = . 2 2 AC BC CD 2 Q16. In  PQR, QPR = 90° and QR = 26 cm. If PS  SR, PS = 6 cm and SR = 8 cm, find ar(  PQR). P S8

cm

Q R 26 cm Q17. A ladder 25 m long reaches a window which is 24 m above the ground on side of the street. Keeping the foot at the same point, the ladder is turned to the other side of the street to reach a window 7 m high. Find the width of the street. Q18. In a quadrilateral ABCD, CA = CD, B = 90° and AD2 = AB2 + BC2 + CA2. Prove that ACD = 90°. Q19. In the given figure, BCA = 90°. Q is the mid-point of BC. Prove that : AB2 = 4AQ2 – 3AC2. A

B

C

Q

Q20. ABC is a right triangle, right angled at B. AD and CE are the two medians drawn from A and C 3 5 respectively. If AC = 5cm and AD = cm , find the length of CE. 2 Q21. ABC is a right triangle, right-angled at C. Let BC = a, CA = b, AB = c and let p be the length of perpendicular from C on AB, prove that: 1 1 1 (i) cp = ab (ii) 2 = 2 + 2 p a b Q22. ABC is a right angles triangle, right angled at A. A circle is inscribed in it. The lengths of the two sides containing the right angles are 6cm and 8cm. Find the radius of the circle. Q23. In the given figure, ABCD is a square. F is the mid-point of AB, BE is one-third of BC. If the area of FBE = 108 cm2, find the length of AC. D

C

A

E B

F

Q24. Equilateral triangles are drawn on the sides of a right angled triangle. Show that the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides. Q25. In ∆ABC, AD is bisector of A, meeting side BC at D. (i) If AB = 10m, AC = 6cm, BC = 12cm find BD and DC. (ii) If AB = 5.6cm, AC = 6cm and DC = 3cm, find BC.