Answer
Verified
368.4k+ views
Hint: Use the fact that the radii of two congruent circles are equal and hence prove that OA = O’A’ and OB = O’B’. Use the fact that since the chords are equal AB = A’B’ and hence prove that the triangle ABC and A’B’C’ are congruent and hence prove that $\angle AOC=\angle A'O'C'$. Hence prove that the equal chords of congruent circles subtend equal angles at the centres of their corresponding circles.
Complete step by step answer:
Given: Two circles with centre O and O’ have equal radii. AB is the chord of the circle with centre O and A’B’ is a chord of the circle with centre O’.
To prove $\angle AOB=\angle A'O'B'$
Proof:
Since the circle have equal radii, we have
OA = O’A’ and OB = O’B’
Now, in triangle AOB and A’O’B’, we have
AO = A’O’ (proved above)
OB = O’B’ (Proved above)
AB = A’B’ (Given).
Hence by S.S.S congruence criterion, we have
$\Delta AOB\cong \Delta A'O'B'$
Hence, we have
$\angle AOB=\angle A'O'B'$ (Corresponding parts of congruent triangles)
Hence, equal chords of two congruent circles subtend equal angles at their respective centres.
Hence, proved.
Note: Alternative Solution: Using Sine rule.
We know that if R is the radius of circumcentre of a triangle ABC, then $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R$
Consider two points C and C’ on the alternate segments as shown in the diagram above.
By sine rule, we have
$\begin{align}
& \dfrac{AB}{\sin C}=2R \\
& \Rightarrow AB=2R\sin C \\
\end{align}$
Similarly, we have
$A'B'=2R\sin C'$
Since AB = A’B’, we have
$\begin{align}
& 2R\sin C=2R\sin C' \\
& \Rightarrow \sin C=\sin C' \\
& \Rightarrow C=C' \\
\end{align}$
We know that the angle subtended in the alternate segment is half the angle subtended at the centre.
Hence, we have
$\angle AOB=2\angle C$ and $\angle A'O'B'=2\angle C'$
Since $\angle C=\angle C'$, we have
$\angle AOB=\angle A'O'B'$
Q.E.D
Complete step by step answer:
Given: Two circles with centre O and O’ have equal radii. AB is the chord of the circle with centre O and A’B’ is a chord of the circle with centre O’.
To prove $\angle AOB=\angle A'O'B'$
Proof:
Since the circle have equal radii, we have
OA = O’A’ and OB = O’B’
Now, in triangle AOB and A’O’B’, we have
AO = A’O’ (proved above)
OB = O’B’ (Proved above)
AB = A’B’ (Given).
Hence by S.S.S congruence criterion, we have
$\Delta AOB\cong \Delta A'O'B'$
Hence, we have
$\angle AOB=\angle A'O'B'$ (Corresponding parts of congruent triangles)
Hence, equal chords of two congruent circles subtend equal angles at their respective centres.
Hence, proved.
Note: Alternative Solution: Using Sine rule.
We know that if R is the radius of circumcentre of a triangle ABC, then $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R$
Consider two points C and C’ on the alternate segments as shown in the diagram above.
By sine rule, we have
$\begin{align}
& \dfrac{AB}{\sin C}=2R \\
& \Rightarrow AB=2R\sin C \\
\end{align}$
Similarly, we have
$A'B'=2R\sin C'$
Since AB = A’B’, we have
$\begin{align}
& 2R\sin C=2R\sin C' \\
& \Rightarrow \sin C=\sin C' \\
& \Rightarrow C=C' \\
\end{align}$
We know that the angle subtended in the alternate segment is half the angle subtended at the centre.
Hence, we have
$\angle AOB=2\angle C$ and $\angle A'O'B'=2\angle C'$
Since $\angle C=\angle C'$, we have
$\angle AOB=\angle A'O'B'$
Q.E.D
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Advantages and disadvantages of science
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
10 examples of evaporation in daily life with explanations
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Difference Between Plant Cell and Animal Cell
What are the monomers and polymers of carbohydrate class 12 chemistry CBSE