## Chapter 4 triangles R.D. Sharma Solutions for Class 10th Math Exercise 4.6

**Exercise 4.6**

1. Triangles ABC and DEF are similar

(i) If area (△ABC) = 16cm

^{2}, area (△DEF) = 25 cm

^{2}and BC = 2.3 cm, find EF.

(ii) If area (△ABC) = 9cm

^{2}, area (△DEF) = 64 cm

^{2}and DE = 5.1 cm, find AB.

(iii)If AC = 19cm and DF = 8 cm, find the ratio of the area of two triangles.

(iv)If area (△ABC) = 36cm

^{2}, area (△DEF) = 64 cm

^{2}and DE = 6.2 cm, find AB.

(v) If AB = 1.2 cm and DE = 1.4 cm, find the ratio of the areas of △ABC and △DEF.

**Solution**

(i)

(ii)

(iii)

(iv)

(v)

2. In fig. below ∆ACB ~ ∆APQ. If BC = 10 cm, PQ = 5 cm, BA = 6.5 cm and AP = 2.8 cm, find CA and AQ. Also, find the area (∆ACB): area (∆APQ).

3. The areas of two similar triangles are 81 cm

5. The areas of two similar triangles are 25 cm

7. ABC is a triangle in which ∠A =90°, AN⊥ BC, BC = 12 cm and AC = 5cm. Find the ratio of the areas of ∆ANC and ∆ABC.

9.In ∆ABC, D and E are the mid-points of AB and AC respectively. Find the ratio of the areas of ∆ADE and ∆ABC .

12. If ∆ABC ~ ∆DEF such that AB = 5 cm, area (∆ABC) = 20 cm

2. In fig. below ∆ACB ~ ∆APQ. If BC = 10 cm, PQ = 5 cm, BA = 6.5 cm and AP = 2.8 cm, find CA and AQ. Also, find the area (∆ACB): area (∆APQ).

**Solution**

^{2}and 49 cm

^{2}respectively. Find the ratio of their corresponding heights. What is the ratio of their corresponding medians ?

**Solution**

4. The areas of two similar triangles are 169 cm

^{2}and 121 cm^{2}respectively. If the longest side
of the larger triangle is 26 cm, find the longest side of the smaller triangle.

**Solution**

^{2}and 36 cm

^{2}respectively. If the altitude of the first triangle is 2.4 cm, find the corresponding altitude of the other.

**Solution**

6. The corresponding altitudes of two similar triangles are 6 cm and 9 cm respectively. Find the ratio of their areas.

**Solution**

7. ABC is a triangle in which ∠A =90°, AN⊥ BC, BC = 12 cm and AC = 5cm. Find the ratio of the areas of ∆ANC and ∆ABC.

**Solution**

8. In Fig. 4.178, DE || BC

(i) If DE = 4 cm, BC = 6 cm and Area (∆ADE) = 16 cm

^{2}, find the area of ∆ABC.
(ii) If DE = 4cm, BC = 8 cm and Area (∆ADE) = 25 cm

^{2}, find the area of ∆ABC.
(iii)If DE : BC = 3 : 5. Calculate the ratio of the areas of ∆ADE and the trapezium BCED .

**Solution**

**Solution**

10. The areas of two similar triangles are 100 cm

^{2}and 49 cm^{2}respectively. If the altitude the
bigger triangle is 5 cm, find the corresponding altitude of the other .

**Solution**

11. The areas of two similar triangles are 121 cm

^{2}and 64 cm^{2}respectively. If the median of the
first triangle is 12.1 cm, find the corresponding median of the other.

**Solution**

12. If ∆ABC ~ ∆DEF such that AB = 5 cm, area (∆ABC) = 20 cm

^{2}and area (∆DEF) = 45 cm^{2}, determine DE.**Solution**

13. In ∆ABC, PQ is a line segment intersecting AB at P and AC at Q such that PQ || BC and PQ divides ∆ABC into two parts equal in area. Find BP/AB .

**Solution**

**Solution**

15. ABC is a triangle and PQ is a straight line meeting AB in P and AC in Q. If AP = 1 cm, PB = 3 cm, AQ = 1.5 cm, QC = 4.5 m, prove that area of ∆APQ is one- sixteenth of the area of ABC.

**Solution**

**Solution**

17. If ∆ABC and ∆BDE are equilateral triangles, where D is the mid-point of BC, find the ratio of areas of ∆ABC and ∆BDE.

**Solution**

**Solution**

**Solution**

20. ABCD is a trapezium in which AB || CD. The diagonals AC and BD intersect at O. Prove

that: (i) ∆AOB and ∆COD (ii) If OA = 6 cm, OC = 8 cm,

Find :

(a) area(∆AOB)area(∆COD)

(b) area(∆AOD)area(∆COD)

**Solution**

**Solution**

22. AD is an altitude of an equilateral triangle ABC. On AD as base, another equilateral triangle ADE is constructed. Prove that Area (∆ADE): Area (∆ABC) = 3: 4.

**Solution**