# In the given figure, find the value of x.

${\text{A}}{\text{. }}{40^0}$

${\text{B}}{\text{. }}{50^0}$

${\text{C}}{\text{. }}{60^0}$

${\text{D}}{\text{. }}{25^0}$

Answer

Verified

382.5k+ views

Hint: Here we go through the properties of the isosceles triangle and the properties of the sum of all internal angles of a triangle which is ${180^0}$ to find the unknown angle.

Given that ABC is a triangle in which D is the midpoint of side BC.

So we can say that BD=DC.

It is also given that the length AD is equal to the length BD and DC.

$\therefore {\text{AD = BD = DC}}$

Now in $\vartriangle {\text{ABD}}$

${\text{AD = BD}}$ (Given in the question)

It means that $\vartriangle {\text{ABD}}$ is an isosceles triangle.

So the base angle of side BD and AD are equal which means $\angle {\text{DBA = }}\angle {\text{DAB = 5}}{{\text{0}}^0}$ given in the figure.

And we know that the sum of all interior angles of a triangle is ${180^0}$.

$

\therefore \angle {\text{DBA + }}\angle {\text{DAB + }}\angle {\text{BDA}} = {180^0} \\

\Rightarrow {50^0} + {50^0} + \angle {\text{BDA}} = {180^0} \\

\Rightarrow \angle {\text{BDA}} = {180^0} - {100^0} \\

\therefore \angle {\text{BDA}} = {80^0} \\

$

And now BDC is a straight line

$\therefore \angle {\text{BDA}} + \angle {\text{ADC = 18}}{{\text{0}}^0}$ Straight angle.

$ \Rightarrow \angle {\text{ADC = 18}}{{\text{0}}^0} - {80^0} = {100^0}$ $\because \angle {\text{BDA}} = {80^0}$

And now in $\vartriangle {\text{ABC}}$

${\text{AD = CD}}$ (Given in question).

It means that $\vartriangle {\text{ADC}}$ is an isosceles triangle.

So the base angle of side BD and AD are equal which means $\angle {\text{DCA = }}\angle {\text{DAC = }}x$ given in the figure.

And we know that the sum of all interior angles of a triangle is ${180^0}$.

$

\therefore \angle {\text{DCA + }}\angle {\text{DAC + }}\angle {\text{ADC}} = {180^0} \\

\Rightarrow x + x + {100^0} = {180^0} \\

\Rightarrow 2x = {180^0} - {100^0} \\

\therefore x = {40^0} \\

$

Hence, option A is the correct answer.

Note: Whenever we face such a type of question the key concept for solving the question is that you have to first read the diagram what it says and the data which are given in the diagrams. Then by applying the properties of triangles start finding the unknown terms to get the answer.

Given that ABC is a triangle in which D is the midpoint of side BC.

So we can say that BD=DC.

It is also given that the length AD is equal to the length BD and DC.

$\therefore {\text{AD = BD = DC}}$

Now in $\vartriangle {\text{ABD}}$

${\text{AD = BD}}$ (Given in the question)

It means that $\vartriangle {\text{ABD}}$ is an isosceles triangle.

So the base angle of side BD and AD are equal which means $\angle {\text{DBA = }}\angle {\text{DAB = 5}}{{\text{0}}^0}$ given in the figure.

And we know that the sum of all interior angles of a triangle is ${180^0}$.

$

\therefore \angle {\text{DBA + }}\angle {\text{DAB + }}\angle {\text{BDA}} = {180^0} \\

\Rightarrow {50^0} + {50^0} + \angle {\text{BDA}} = {180^0} \\

\Rightarrow \angle {\text{BDA}} = {180^0} - {100^0} \\

\therefore \angle {\text{BDA}} = {80^0} \\

$

And now BDC is a straight line

$\therefore \angle {\text{BDA}} + \angle {\text{ADC = 18}}{{\text{0}}^0}$ Straight angle.

$ \Rightarrow \angle {\text{ADC = 18}}{{\text{0}}^0} - {80^0} = {100^0}$ $\because \angle {\text{BDA}} = {80^0}$

And now in $\vartriangle {\text{ABC}}$

${\text{AD = CD}}$ (Given in question).

It means that $\vartriangle {\text{ADC}}$ is an isosceles triangle.

So the base angle of side BD and AD are equal which means $\angle {\text{DCA = }}\angle {\text{DAC = }}x$ given in the figure.

And we know that the sum of all interior angles of a triangle is ${180^0}$.

$

\therefore \angle {\text{DCA + }}\angle {\text{DAC + }}\angle {\text{ADC}} = {180^0} \\

\Rightarrow x + x + {100^0} = {180^0} \\

\Rightarrow 2x = {180^0} - {100^0} \\

\therefore x = {40^0} \\

$

Hence, option A is the correct answer.

Note: Whenever we face such a type of question the key concept for solving the question is that you have to first read the diagram what it says and the data which are given in the diagrams. Then by applying the properties of triangles start finding the unknown terms to get the answer.

Recently Updated Pages

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

Which country launched the first satellite in space class 11 physics CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is the past tense of read class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What is pollution? How many types of pollution? Define it

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE