CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

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


Solution

Given : 
A quadrilateral $$PQRS$$ circumscribes a circle with center $$O$$. Sides of quadrilateral $$PQ, QR, RS$$ and $$SP$$ touches the circle at point $$L, M, N, T$$ respectively.
To prove : $$\angle POQ + \angle SOR = 180^{\circ}$$
and $$\angle SOP + \angle ROQ = 180^{\circ}$$
Construction : Join $$P, Q, R, L, M, N$$ and $$T$$ with center $$O$$ of circle,
Proof : Since, $$OL, OM, ON$$ and $$OT$$ are radius of circle and $$QL, MQ, RN$$ and $$ST$$ are tangents of circle. So
$$QL \perp OL \perp QM \perp OM, RN \perp ON$$ and $$ST \perp OT$$ 
Now in right angled $$\Delta OMQ$$ and right $$\Delta OLQ$$
$$\angle OMQ = \angle OLQ$$ ( each  $$90^{\circ}$$)
hypotenuse $$OQ$$ = hypotenuse $$OQ$$ (common side)
and $$OM = OL$$ (equal radii of circle)
$$\therefore OMQ = OLQ$$ (By $$RHS$$ Congruence)
$$\Rightarrow \angle 3 = \angle 2 (CPCT)$$
Similarly 
$$\angle 4 = \angle 5$$
$$\angle 6 = \angle 7$$ and $$\angle 8 = \angle 1$$
$$\because$$ Sum of all angles made on point $$O$$ of center of circle $$= 360^{\circ}$$
$$\therefore \angle 1 + \angle 2 + \angle 3 + \angle 4 + \angle 5 + \angle 6 + \angle 7 + \angle 8 = 360^{\circ}$$
$$\Rightarrow  \angle 1 + \angle 2 + \angle 2 + \angle 5 + \angle 5 + \angle 6 + \angle 6 + \angle 1 = 360^{\circ}$$
$$\Rightarrow  2(\angle 1 + \angle 2 + \angle 5 + \angle 6) = 360^{\circ}$$
$$\Rightarrow  (\angle 1 + \angle 2) + (\angle 5 + \angle 6) = 180^{\circ}$$
$$\angle POQ + \angle SOR = 180^{\circ}$$
$$[\because \angle 1 + \angle 2 = \angle POQ$$ and $$\angle 5 + \angle 6 = \angle SOR]$$
$$\angle SOP + \angle ROQ = 180^{\circ}$$
So, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of circle.

1863938_1876531_ans_c4f9a076328b4e5dabf34b7f844d7115.png

Mathematics

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
People also searched for
View More



footer-image