Ex.10.2 Q13 Circles Solution - NCERT Maths Class 10

Go back to  'Ex.10.2'

Question

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.

 

Text Solution

  

To prove:

Opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.

Steps and Reasoning:

We know that, tangents drawn from an external point to a circle subtend equal angles at the center.

In the above figure, \({P,\;Q,\;R,\;S}\) are point of contacts

\({AS = AP}\) (The lengths of tangents drawn from an external point \(A\) to a circle are equal.)

\[\begin{align} \angle {SOA} = \angle {POA} = \angle 1 = \angle 2 \end{align}\]

Tangents drawn from an external point to a circle subtend equal angles at the centre.

Similarly,

\[\begin{align} \angle 3 = \angle 4 , \angle 5 = \angle 6 , \angle 7 = \angle 8 \end{align}\]

Since complete angle is \({360^ \circ }\),

\[\begin{align} \angle 1+\angle 2+\angle 3+\angle 4+\angle 5+\angle 6+\angle 7+\angle 8 &=360^{\circ} \\ 2(\angle 1+\angle 8+\angle 4+\angle 5) &=360^{\circ} \\ \angle 1+\angle 8+\angle 4+\angle 5 &=180^{\circ} \end{align}\]

[or]

\[\begin{align} 2(\angle 2+\angle 3+\angle 6+\angle 7) &=360^{\circ} \\ \angle 2+\angle 3+\angle 6+\angle 7 &=180^{\circ} \\ \angle A O B+\angle C O D &=180^{\circ} \end{align}\]

From above figure \(\begin{align}\angle A O D+\angle B O C =180^{\circ}\end{align}\)

\(\angle {AOD}\) and \(\angle {BOC}\) are angles subtended by opposite sides of quadrilateral circumscribing a circle and sum of them is \({180^ \circ }\)

Hence proved.

  

Frequently Asked Questions



What are Class 10 NCERT Exemplars?
While getting good scores in school tests is a desirable outcome, it is not a reliable indicator of how strong your child’s math foundation really is. Many students who score well in school exams in their earlier years, might struggle with math in higher grades because of a weak foundation. At Cuemath, we evaluate your child’s grasp of math fundamentals, and take corrective actions immediately. Also, your child may have limited exposure in their school, and in most cases, may not feel challenged to learn more. Cuemath's customised learning plan ensures your child is challenged with varied difficulty levels of questions at every stage.
What is the difference between CBSE and NCERT syllabus for Class 10?
How will Class 10 NCERT books help in exam preparation?
How will Class 10 NCERT books help you understand basic math concepts?
Which is the best video solution for the class 10 maths NCERT?