Answer
Verified
396.9k+ views
Hint: In this question, we will first draw the diagram. After this, we use the concept that the radius of the circle is equal, we will find MB and NC. Now, we will apply the SAS congruence rule in triangle MBC and triangle NBC.
Complete step-by-step answer:
Let us first draw the diagram of the question:
A is the centre of the circle and AM and AN are the radius of the circle.
$ \therefore $ AM = AN (1)
It is given that triangle ABC is an equilateral triangle. So, AB = BC = CA and $ \angle $ MBC = $ \angle $ NCB = $ \angle $ BAC = $ {60^0} $
Also, AM+MB = AN+NC
From equation 1, we get:
MB = NC.
Now, in $ \vartriangle MBC{\text{ and }}\vartriangle {\text{NBC}} $ , we have:
MB = NC
$ \angle $ MBC= $ \angle $ NCB = $ {60^0} $
And BC = BC
SO, by SAS rule of congruence, we can say that:
$ \vartriangle MBC \cong \vartriangle NBC $
And by c.p.c.t., we can say that CM = BN.
Therefore, we have proved that CM= BN.
Note: In this type of question, you should first draw the diagram so that things gets clear for further calculation. You should remember the different rules of congruency which are given below:
1. SSS rule – All sides of one triangle are equal to all corresponding sides of another triangle.
2.SAS- Two sides and one angle formed between these two sides of one triangle is equal to that other triangle.
3. ASA- one side and two angles formed on this side of one triangle is equal to that of the other triangle.
4. RHS- This is applicable for the right triangle. Hypotenuse and one other side of one triangle is equal to that of another triangle.
Complete step-by-step answer:
Let us first draw the diagram of the question:
A is the centre of the circle and AM and AN are the radius of the circle.
$ \therefore $ AM = AN (1)
It is given that triangle ABC is an equilateral triangle. So, AB = BC = CA and $ \angle $ MBC = $ \angle $ NCB = $ \angle $ BAC = $ {60^0} $
Also, AM+MB = AN+NC
From equation 1, we get:
MB = NC.
Now, in $ \vartriangle MBC{\text{ and }}\vartriangle {\text{NBC}} $ , we have:
MB = NC
$ \angle $ MBC= $ \angle $ NCB = $ {60^0} $
And BC = BC
SO, by SAS rule of congruence, we can say that:
$ \vartriangle MBC \cong \vartriangle NBC $
And by c.p.c.t., we can say that CM = BN.
Therefore, we have proved that CM= BN.
Note: In this type of question, you should first draw the diagram so that things gets clear for further calculation. You should remember the different rules of congruency which are given below:
1. SSS rule – All sides of one triangle are equal to all corresponding sides of another triangle.
2.SAS- Two sides and one angle formed between these two sides of one triangle is equal to that other triangle.
3. ASA- one side and two angles formed on this side of one triangle is equal to that of the other triangle.
4. RHS- This is applicable for the right triangle. Hypotenuse and one other side of one triangle is equal to that of another triangle.
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
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
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Summary of the poem Where the Mind is Without Fear class 8 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Write an application to the principal requesting five class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE