Answer
450.9k+ views
Hint: Consider, \[\vartriangle PQR\], use the properties of the isosceles triangle and find the \[\angle PRQ\]. PR and QR are tangents from a point R meeting the circle at P and Q.
“Complete step-by-step answer:”
Given a figure, with a circle whose center is O. From the figure we can say that PR and QR are tangents to the circle from a point R. Three tangents from an external point to the circle are equal.
\[\therefore PR=QR\]
Now let us consider the isosceles triangle, \[\vartriangle PRQ\].
As it is an isosceles triangle, we know that two sides of an isosceles triangle are equal.
In \[\vartriangle PRQ\], we know that PR = QR.
Thus the angle RPQ is equal to angle PQR, because they are the base angles of an isosceles triangle.
We have been given, \[\angle RPQ={{45}^{\circ }}\].
\[\angle RPQ=\angle PQR={{45}^{\circ }}\].
The angle sum property of triangle, states that the sum of interior angles of a triangle is \[{{180}^{\circ }}\].
Hence, in the isosceles triangle PRQ, the sum of all the interior angles is \[{{180}^{\circ }}\].
\[\therefore \angle RPQ+\angle PQR+\angle PRQ={{180}^{\circ }}\]
We need to find the \[\angle PRQ\].
We already found out that, \[\angle RPQ=\angle PQR={{45}^{\circ }}\].
Thus substituting these values in the sum of triangle,
\[\begin{align}
& \angle RPQ+\angle PQR+\angle PRQ={{180}^{\circ }} \\
& {{45}^{\circ }}+{{45}^{\circ }}+\angle PRQ={{180}^{\circ }} \\
& \angle PRQ=180-45-45=180-90 \\
& \angle PRQ={{90}^{\circ }} \\
\end{align}\]
Hence, we got \[\angle PRQ={{90}^{\circ }}\], which makes \[\vartriangle PRQ\] right angled at R.
\[\therefore \]Option (b) is the correct answer.
Note: By seeing the figure, you may try to take \[\vartriangle POQ\] instead of \[\vartriangle PRQ\], understand the question that we only need to find \[\angle PRQ\]. So we should take, \[\vartriangle PRQ\]. As tangents from the external point of the circle are equal, the sides of the triangle are same, thus angles are same and we can get \[\angle PRQ\]easily.
“Complete step-by-step answer:”
Given a figure, with a circle whose center is O. From the figure we can say that PR and QR are tangents to the circle from a point R. Three tangents from an external point to the circle are equal.
\[\therefore PR=QR\]
Now let us consider the isosceles triangle, \[\vartriangle PRQ\].
As it is an isosceles triangle, we know that two sides of an isosceles triangle are equal.
In \[\vartriangle PRQ\], we know that PR = QR.
Thus the angle RPQ is equal to angle PQR, because they are the base angles of an isosceles triangle.
We have been given, \[\angle RPQ={{45}^{\circ }}\].
\[\angle RPQ=\angle PQR={{45}^{\circ }}\].
The angle sum property of triangle, states that the sum of interior angles of a triangle is \[{{180}^{\circ }}\].
Hence, in the isosceles triangle PRQ, the sum of all the interior angles is \[{{180}^{\circ }}\].
\[\therefore \angle RPQ+\angle PQR+\angle PRQ={{180}^{\circ }}\]
We need to find the \[\angle PRQ\].
We already found out that, \[\angle RPQ=\angle PQR={{45}^{\circ }}\].
Thus substituting these values in the sum of triangle,
\[\begin{align}
& \angle RPQ+\angle PQR+\angle PRQ={{180}^{\circ }} \\
& {{45}^{\circ }}+{{45}^{\circ }}+\angle PRQ={{180}^{\circ }} \\
& \angle PRQ=180-45-45=180-90 \\
& \angle PRQ={{90}^{\circ }} \\
\end{align}\]
Hence, we got \[\angle PRQ={{90}^{\circ }}\], which makes \[\vartriangle PRQ\] right angled at R.
\[\therefore \]Option (b) is the correct answer.
Note: By seeing the figure, you may try to take \[\vartriangle POQ\] instead of \[\vartriangle PRQ\], understand the question that we only need to find \[\angle PRQ\]. So we should take, \[\vartriangle PRQ\]. As tangents from the external point of the circle are equal, the sides of the triangle are same, thus angles are same and we can get \[\angle PRQ\]easily.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)