  ∠ 1 + ∠ 2 + ∠ 5 + ∠ 6 = (360°)/2 Now The area can be divided into four kites. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. Hence proved. AP = AS (∠ 1 + ∠ 2) + (∠ 5 + ∠ 6) = 180° Sabu. Find the sides AB and AC. He has been teaching from the past 9 years. It means that all the four vertices of quadrilateral lie in the circumference of the circle. As, sum of adjacent angles is supplementary (180°), hence opposite sides are parallel. Given : Let ABCD be the quadrilateral circumscribing the circle with centre O. Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. There is two important theorems which prove the cyclic quadrilateral. Join AO, BO, CO, DO. Let ABCD be a quadrilateral circumscribing a circle with centre O. From a tangential quadrilateral, one can form a hexagon with two 180° angles, by placing two new vertices at two opposite points of tangency; all six of the sides of this hexagon lie on lines tangent to the inscribed circle, so its diagonals meet at a point. In our case the sums of the opposite sides are of 5 + 7 = 12 cm A quadrilateral is drawn to circumscribe a circle. Ans. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. Solution: AB touches at P. There are two theorems by Ptolemy about the diagonals of cyclic quadrilaterals that enable a direct solution to this problem. Teachoo is free. The definition states that a quadrilateral which circumscribed in a circle is called a cyclic quadrilateral. Mar 12, 2015. ∠ AOP = ∠ AOS Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. Solution: P is the point of contact of tangent line l. Let, OP⊥ l at Point of contact P and it passes through point O. The angles BAO and DAO formed at A are equal, as AB and AD are tangents. ∴ Δ AOP ≅∆ AOS Problem PG-010 The quadrilateral ABCD shown in Fig. Find the length of BC. In our case the sums of the opposite sides are of 5 + 7 = 12 cm and 6 + 8 = 14 cm. Prove that this quadrilateral is not circumscribed about a circle. ∠ 1 + ∠ 2 + ∠ 2 + ∠ 5 + ∠ 5 + ∠ 6 + ∠ 6 + ∠ 1 = 360° Given : Let A.. In figure, a triangle ABC is drawn to circumscribe a circle of radius 4 cm, such that the segments BD and DC are of lengths 8 cm and 6 cm respectively. Similarly, we can prove that ∠BOC + ∠DOA = 180º Hence, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. When a quadrilateral in inscribed in a circle, it is called a cyclic quadrilateral. Steps and Reasoning: We know that, tangents drawn from an external point to a circle subtend equal angles at the center. Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. Prove that the circle drawn on any one of the equal sides of an isosceles triangles as diameter bisects the base. There is two important theorems which prove the cyclic quadrilateral. 10.14). Solution: Consider the following figure: Focus your attention on $$\Delta {\rm{AOE}}$$. 86.9k VIEWS. ∠6 = ∠7 Question 5. 2 (∠ 1 + ∠ 2 + ∠ 5 + ∠ 6) = 360° Prove that the sums of opposite sides are equal. Thus, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. 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