Loading web-font TeX/Math/Italic
Home CBSE 10th Class MATHEMATICS (10th)

Exercise 6.4

Exercise 6.4

1.Let ∆ ABC ~ ∆ DEF and their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC.

Sol. ∆ ABC ~ ∆ DEF      |Given

\therefore \frac{ar(∆ ABC)}{ar(∆ DEF)} = (\frac{BC}{EF})^{2}

|\because

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides

Putting given values,

\frac{64}{121} = (\frac{BC}{15.4})^{2} \Rightarrow (\frac{8}{11})^{2} = (\frac{BC}{15.4})^{2}

\Rightarrow \frac{8}{11} = \frac{BC}{15.4}

       |Taking square root on both sides

Cross – multiplying

11 X BC = 8 X 15.4

\Rightarrow BC = \frac{8 X 15.4}{11}

= 8 X 1.4

\Rightarrow

BC = 11.2 cm.

2.Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2 CD, find the ratio of the areas of triangles AOB and COD.

Sol.In ∆ AOB and ∆ COD,

\angle AOB = \angle COD

     |Vert.opp.\angle S

\angle OAB = \angle OCD

    |Alt. Int. \angle S

[\because

AB || DC (given)]

\therefore

∆ AOB ~ ∆ COD

|AA similarity criterion

\therefore \frac{ar(∆ AOB}{ar(∆ COD)} = (\frac{AB}{CD})^{2}

|\because

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides

= (\frac{2CD}{CD})^{2}

       |\because
AB = 2CD (given)

= (\frac{2}{1})^{2} = \frac{4}{1}

= 4 : 1.

3.In Fig. 6.44, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that \frac{ar(ABC)}{ar(DBC)} = \frac{AO}{DO}

.

Sol.Given : In figure, ABC and DBC are two triangles on the same base BC and AD intersects BC at O.

To Prove : \frac{ar(∆ ABC)}{ar(∆ DBC)} = \frac{AO}{DO}

Construction : Draw AP \perp BC

and DQ \perp BC
.

Proof : In ∆ AOP and ∆ DOQ

\angle APO = \angle DQO

|Each equal to 90^{0}

(By construction)

\angle AOP = \angle DOQ

    |vert.opp.\angle S

\therefore

∆ AOP ~ ∆ DOQ

|AA similarity criterion

\therefore \frac{AP}{DQ} = \frac{AO}{DO}

      … (1)

|\because

Corresponding sides of two similar triangles are proportional

Now, \frac{ar(∆ ABC}{ar(∆ DBC} = \frac{\frac{1}{2}(BC)(AP)}{\frac{1}{2}(BC)(DQ)}

= \frac{AP}{DQ}

= \frac{AO}{DO}

     |From ....     (1)

4.If the areas of two similar triangles are equal, prove that they are congruent.

Sol.Given : ∆ ABC and ∆ DEF are two similar triangle such that

ar(∆ ABC) = ar(∆ DEF)

To Prove : ∆ ABC \cong ∆ DEF

Proof : ∆ ABC ~ ∆ DEF      |Given

\therefore \frac{ar(∆ ABC)}{ar(∆ DEF)} = (\frac{BC}{EF})^{2}

|\therefore

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides

But ar (∆ ABC) = ar (∆ DEF) (given)

\Rightarrow 1 = (\frac{BC}{EF})^{2}

Taking square roots,

1 = \frac{BC}{EF}

\Rightarrow

BC = EF      … (1)

Also, \angle B = \angle E

      …. (2)

and \angle C = \angle F

   … (3)

|\because

∆ ABC ~ ∆ DEF   (given)

In view of (1), (2) and (3),

∆ ABC \cong ∆ DEF

|ASA congruence criterion

5.D, E and F are respectively the mid-points of sides AB, BC and CA of ∆ ABC. Find the ratio of the areas of ∆ DEF and ∆ ABC.

Sol.Given :D, E and F are respectively the mid-points of sides AB, BC and CA of ∆ ABC.

To Determine. Ratio of the areas of ∆ DEF and ∆ ABC.

Determination.

Because D and E are mid – points of sides AB and BC respectively  of ∆ ABC (given),

\therefore

DE || AC i.e., DE || AF

Similarly EF || AD

\therefore

Quadrilateral ADEF is a parallelogram

\angle DEF = \angle A

[\because

Opposite angles of a parallelogram are equal]

Arguing as above, BEFD is also a parallelogram.

\therefore \angle DFE = \angle B

    … (2)

[\because

Opposite angles of a parallelogram are equal]

In view of (1) and (2),

∆ DEF ~ ∆ ABC

|AA similarity criterion

\therefore \frac{ar(∆ DEF)}{ar(∆ ABC)} = (\frac{DF}{BC})^{2}

|\because

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides

= \left ( \frac{\frac{1}{2}BC}{BC} \right )^{2}

|\because

D and F are the mid-points of AB and AC respectively \therefore
DF || BC   and DF = \frac{1}{2} BC

= (\frac{1}{2})^{2} = \frac{1}{4}

\therefore

ar (∆ DEF): ar (∆ ABC) = 1 : 4.

Remark. \therefore

ar (∆ ABC) : 4 ar (∆ DEF)

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

Sol.Given : Two triangles ABC and DEF such that

∆ ABC ~ ∆ DEF

Let AM and DN be their corresponding medians.

To Prove : \frac{ar(∆ ABC)}{ar(∆ DEF)} = (\frac{AM}{DN})^{2}

Proof : ∆ ABC ~ ∆ DEF        |Given

\therefore \frac{ar(∆ ABC)}{ar(∆ DEF)} = (\frac{AB}{DE})^{2}

      ….. (1)

|\because

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides and \frac{AB}{DE} = \frac{BC}{EF}

|\because

The corresponding sides of two similar triangles are proportional

\Rightarrow \frac{AB}{DE} = \frac{2BM}{2EN}

|\because

AM and DN are the medians \Rightarrow
M is mid-point of BC and N is mid-point of EF.

= \frac{BM}{EN}

   …. (2)

Also, \angle ABM = \angle DEN

    …. (3)

|\because

∆ ABC ~ ∆ DEF

\therefore \angle ABC = \angle DEF

(Corresponding angles of two similar triangles are equal)

In view of (2) and (3),

∆ ABM ~ ∆ DEN

|SAS similarity criterion

\therefore \frac{AB}{DE} = \frac{BM}{EN} = \frac{AM}{DN}

    …. (4)

|\because

Corresponding sides of two similar triangles are proportional

Putting \frac{AB}{DE} = \frac{AM}{DN}

from (4) in (1), we have

\frac{ar(∆ ABC)}{ar(∆ DEF)} = (\frac{AM}{DN})^{2}

.

7.Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.

Sol.Given : ABCD is a square whose one diagonal is AC.

∆ BQC and ∆ APC are two equilateral triangles described on the side BC and diagonal AC of the square ABCD.

To Prove : ar (∆ BQC) = \frac{1}{2}

ar(∆ APC)

Proof : \because

∆ APC and ∆ BQC are both equilateral triangles

\therefore

∆ APC ~ ∆ BQC

|AAA similarity criterion

\therefore \frac{ar(∆ APC)}{ar(∆ BQC)} = (\frac{AC}{BC})^{2}

|\therefore

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides

= (\frac{\sqrt{2}BC}{BC})^{2}

|\because

Diagonal of square = \sqrt{2}
side

= (\sqrt{2})^{2}

= 2

\Rightarrow \frac{ar(∆ APC)}{ar(∆ BQC)}

= 2

Cross – multiplying

2 ar (∆ BQC) = ar (∆ APC)

\Rightarrow ar (∆ BQC) = \frac{1}{2} ar(∆ APC)

.

TICK the correct answer and justify :

8.ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is

(A)2 : 1      (B)1 : 2         (C)4 : 1          (D)1 : 4.

Sol.\because

∆ ABC and ∆ BDE are both equilateral triangles

\therefore

∆ ABC ~ ∆ BDE

|AAA similarity criterion

\therefore \frac{ar(∆ ABC)}{ar(∆ BDE)} = (\frac{AB}{BD})^{2}

|\because

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides

= (\frac{BC}{BD})^{2}

        |\because
AB = BC = CA for equilateral ∆ ABC

= (\frac{2BD}{BD})^{2}

|\because

D is the mid – point of BC

= \frac{4}{1}

\Rightarrow

ar (∆ ABC) : ar (∆ BDE) = 4 : 1

Hence, (C) 4 : 1 is the correct answer.

9.Sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio

(A)2 : 3         (B)4 : 9            (C)81 : 16           (D)16 : 81

Sol.Ratio of the areas of these triangles = 4^{2} : 9^{2}

|\because

The ratio of the similar triangles is equal to the square of the ratio of their corresponding sides

Hence, (D) 16 : 81 is the correct answer.