Vardaan Learning Institute

1. If $\triangle ABC \sim \triangle PQR$, area($\triangle ABC$) = 81 $cm^2$ and area($\triangle PQR$) = 144 $cm^2$. If AB = 9 cm, then PQ is:

1. 12 cm
2. 16 cm
3. 4 cm
4. 18 cm

2. In $\triangle ABC$, DE || BC. If AD = 1.5 cm, DB = 3 cm and AE = 1 cm, then EC is:

1. 1.5 cm
2. 3 cm
3. 2 cm
4. 4 cm

3. The lengths of the diagonals of a rhombus are 24 cm and 32 cm. The side of the rhombus is:

1. 12 cm
2. 16 cm
3. 20 cm
4. 30 cm

4. Two poles of height 6 m and 11 m stand vertically upright on a plane ground. If the distance between their feet is 12 m, the distance between their tops is:

1. 13 m
2. 14 m
3. 15 m
4. 12 m

5. A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. The height of the tower is:

1. 40 m
2. 42 m
3. 38 m
4. 44 m

6. Two isosceles triangles have equal angles and their areas are in the ratio 16:25. The ratio of their corresponding heights is:

1. 4:5
2. 5:4
3. 3:2
4. 5:7

7. If $\triangle ABC \sim \triangle DEF$, BC = 3 cm, EF = 4 cm and area of $\triangle ABC = 54 cm^2$. Then area of $\triangle DEF$ is:

1. 96 $cm^2$
2. 100 $cm^2$
3. 72 $cm^2$
4. 48 $cm^2$

8. A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall.

1. 6 m
2. 8 m
3. 4 m
4. 5 m

9. In $\triangle ABC$, if $AB^2 = AC^2 + BC^2$, then the angle opposite to side AB is:

1. 45 deg
2. 60 deg
3. 90 deg
4. 30 deg

10. Assertion (A): Two similar triangles are always congruent.
Reason (R): If the areas of two similar triangles are equal, then they are congruent.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is not the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

11. In $\triangle ABC$, if angles P, Q, R are mid-points of the sides BC, CA and AB respectively. Find the ratio of the area of $\triangle PQR$ to area of $\triangle ABC$.

12. In a trapezium ABCD, AB || DC and its diagonals intersect each other at the point O. Show that $AO/BO = CO/DO$.

13. The perimeters of two similar triangles ABC and PQR are 60 cm and 48 cm respectively. If PQ = 8 cm, find AB.

14. In the given figure:
If $\triangle ABE \cong \triangle ACD$, show that $\triangle ADE \sim \triangle ABC$.

15. D is a point on the side BC of a triangle ABC such that $\angle ADC = \angle BAC$. Show that $CA^2 = CB \cdot CD$.

16. Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of $\triangle PQR$. Show that $\triangle ABC \sim \triangle PQR$.

17. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.

18. BL and CM are medians of a triangle ABC right angled at A. Prove that $4(BL^2 + CM^2) = 5 BC^2$.

19. Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting AC in L and AD produced in E. Prove that $EL = 2 BL$.

20. A vertical stick 12 m long casts a shadow 8 m long on the ground. At the same time a tower casts the shadow 40 m long on the ground. Find the height of the tower.

21. State and prove the Basic Proportionality Theorem (Thales Theorem).

22. State and prove the Pythagoras Theorem.

23. Case Study: Light House
A boy of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground:

(i) Draw a rough sketch of the situation. (1 Mark)
(ii) Find the length of his shadow after 4 seconds. (2 Marks)
(iii) Which similarity criterion is used to solve this problem? (1 Mark)