Prove that opposite sides of a quadrilateral circumscribing a circle, subtend supplementary ang... Video Lecture - Class 10

Ans. A quadrilateral circumscribing a circle is a quadrilateral in which all four sides are tangents to the circle. This means that the circle is inscribed within the quadrilateral such that each side of the quadrilateral touches the circle at a single point.

Ans. When opposite sides of a quadrilateral subtend supplementary angles, it means that the sum of the measures of the angles formed by the intersection of the opposite sides is equal to 180 degrees. In other words, if we denote the angles as A, B, C, and D, where A and C are opposite angles and B and D are opposite angles, then A + C = 180 degrees and B + D = 180 degrees.

Ans. To prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles, we can use the fact that the angle between a tangent and a chord at the point of contact is equal to the angle subtended by the chord in the opposite segment. Here's a step-by-step proof: 1. Let ABCD be a quadrilateral circumscribing a circle, with the circle centered at O. 2. Let the tangents from A and C intersect at point P, and the tangents from B and D intersect at point Q. 3. From the tangent-chord angle theorem, we know that angle BAP = angle BCO and angle ADP = angle ADO. 4. Similarly, angle CDP = angle CBO and angle ABQ = angle ACQ. 5. Since angles BAP and ACQ are vertically opposite angles, they are equal. Similarly, angles ADP and ABQ are equal. 6. Therefore, angle BAP + angle ACQ = angle ADP + angle ABQ. 7. This can be rewritten as angle BCO + angle CBO = angle ADO + angle OAB. 8. Simplifying further, we get angle BCO + angle CBO + angle ADO + angle OAB = 180 degrees. 9. Hence, the opposite angles A and C subtend supplementary angles, as do the opposite angles B and D.

Ans. A quadrilateral circumscribing a circle has several significant properties. One important property is that the sums of opposite angles are supplementary, as proven earlier. This property is used in many geometric proofs and constructions. Additionally, the fact that all four sides of the quadrilateral are tangents to the circle means that the lengths of the tangents from a point outside the circle to the circle are equal. This property is often used in solving problems involving tangents and circles. Furthermore, the circle inscribed within the quadrilateral can also be used to find the area of the quadrilateral using the formula A = rs, where A is the area, r is the radius of the inscribed circle, and s is the semiperimeter of the quadrilateral.

Ans. Some examples of quadrilaterals circumscribing a circle include: - Square: All four sides of a square are tangents to the inscribed circle. - Rectangle: If the length and width of a rectangle are equal, it becomes a square and also circumscribes a circle. - Rhombus: A rhombus with equal diagonals circumscribes a circle. - Trapezoid: In a trapezoid, the sum of the lengths of the bases is equal to the sum of the lengths of the other two sides, and it circumscribes a circle. - Kite: A kite with perpendicular diagonals circumscribes a circle.