## Chapter 4 triangles R.D. Sharma Solutions for Class 10th Math Exercise 4.5

**Exercise 4.5**

1. In fig. 4.136, âˆ†ACB ~ âˆ†APQ. If BC = 8 cm, PQ = 4 cm, BA = 6.5 cm and AP = 2.8 cm, find CA andÂ AQ.

**Solution**

2. In Fig. 4.137, AB || QR. Find the length of PB.Â

**Solution**

3. In fig. 4.138, XY || BC. Find the length of XY .

**Solution**

4. In a right angled triangle with sides a and b and hypotenuse c, the altitude drawn on the hypotenuse is x. Prove that ab = cx.

**Solution**

5. In Fig. 4.139, âˆ ABC = 90Â° and BD âŠ¥ AC. If BD = 8 cm and AD = 4 cm, find CD .

**Solution**

6. In Fig. 4.140, âˆ ABC = 90Â° and BD âŠ¥ AC. If AB = 5.7 cm, BD = 3.8 cm and CD = 5.4 cm, find BC.

**Solution**

7. In Fig. 4.141, DE || BC such that AE = (1/4) AC. If AB = 6 cm, find AD .

**Solution**

8. In fig. 4.412, if AB âŸ‚ BC, DC âŸ‚ BC and DE âŸ‚ AC, prove that â–³CED ~ â–³ABC.

**Solution**

Given,

AB âŸ‚ BC

DC âŸ‚ BC

So, Both AB and DC are perpendicular to same line BC.

We know that, two lines, perpendicular to same line are parallel.

âˆ´ AB | | DC

Since, AC is transverse (from diagram) (i)

Now, In â–³ABC and â–³DCE

âˆ ABC = âˆ DCE = 90Â° (Given ABâŸ‚BC)

âˆ BAC = âˆ DCE (from i)

Therefore, by AA criteria for similarity

â–³CED ~ â–³ABC .

9. Diagonals AC and BD of trapezium ABCD with AB || DC intersect each other at the point O. Using similarity criterion for two triangles, show that OA/OC = OB/OD .

**Solution**

It is given that trapezium ABCD with AB | | DC.O is the point of intersection of AC and BD .

We have to prove that OA/OC = OB/OD

Now, in â–³AOB and â–³COD

âˆ AOB = âˆ COD (Vertically opposite angles)

âˆ OAB = âˆ OCD (Alternate angles)

âˆ´ â–³AOB ~ â–³COD (AA Similarity)

Hence, OA/OC = OB/OD (Corresponding sides are proportional)

10. If â–³ABC and â–³AMP are two right triangles, triangles, right angled at B and M respectively such that âˆ MAP = âˆ BAC. Prove that

(i) â–³ABC ~ â–³AMP

(ii) CA/PA = BC/MP

**Solution**

(i) It is given that â–³ABC and â–³AMP are two right angle triangles,

Now, in â–³ABC and â–³AMP, we have

âˆ MAP = âˆ BAC (Given)

âˆ AMP = âˆ B = 90Â°

â–³ABC ~ â–³AMP (AA Similarity)

(ii) â–³ABC ~ â–³AMP

So, CA/PA = BC = MP (corresponding sides are proportional) .

11. A vertical stick 10 cm long casts a shadow 8 cm long. At the same time a shadow 30 m long. Determine the height of the tower.

**Solution**

12. In below fig., âˆ A = âˆ CED, Prove that âˆ†CAB ~ âˆ†CED. Also, find the value of x.

**Solution**

13. The perimeters of two similar triangles are 25 cm and 15 cm respectively. If one side of first triangle is 9 cm, what is the corresponding side of the other triangle?

**Solution**

14. In âˆ†ABC and âˆ†DEF, it is being given that: AB = 5 cm, BC = 4 cm and CA = 4.2 cm; DE =10cm, EF = 8 cm and FD = 8.4 cm. If AL âŠ¥ BC and DM âŠ¥ EF, find AL: DM.

**Solution**

15. D and E are the points on the sides AB and AC respectively of a âˆ†ABC such that: AD = 8 cm, DB = 12 cm, AE = 6 cm and CE = 9 cm. Prove that BC = 5/2 DE.

**Solution**

16. D is the mid-point of side BC of a âˆ†ABC. AD is bisected at the point E and BE produced cuts AC at the point X. Prove that BE : EX = 3 : 1.

**Solution**

17. ABCD is a parallelogram and APQ is a straight line meeting BC at P and DC produced at Q. Prove that the rectangle obtained by BP and DQ is equal to the AB and BC.

**Solution**

18. In â–³ABC, AL and CM are the perpendiculars from the vertices A and C to BC and AB respectively . AL and CM intersect at O, prove that :

(i) â–³OMA and â–³OLC

(ii) OA/OC = OM/OL

**Solution**

19. ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the mid-points of AB, AC, CD and BD respectively, show that PQRS is a rhombus.

**Solution**

20. In an isosceles âˆ†ABC, the base AB is produced both the ways to P and Q such that AP Ã— BQ = AC

^{2}. Prove that â–³APC ~ â–³BCQ .

**Solution**

21. A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2m/sec. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds.

**Solution**

22. A vertical stick 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. Find the height of the tower.

**Solution**

23. In below Fig., âˆ†ABC is right angled at C and DE âŠ¥ AB. Prove that âˆ†ABC ~ âˆ†ADE and Hence, find the lengths of AE and DE .

**Solution**

24. In fig., 4.142, PA, QB and RC are each perpendicular to AC. Prove that 1/x + 1/z + 1/y .

**Solution**

25. In Fig below we have AB || CD || EF. If AB = 6 cm, CD = x cm, EF = 10 cm, BD = 4 cm and DE = y cm, calculate the values of x and y.

**Solution**