Cyclic Quadrilateral Test (Motivate) MCQs Quiz | Class 9
Practice this Class IX Mathematics (Code 041) Unit IV Geometry quiz on the topic Cyclic Quadrilateral Test (Motivate) MCQs Quiz. These questions cover key concepts regarding properties of cyclic quadrilaterals and the condition where equal angles subtended by a line segment at two distinct points on the same side imply concyclic points. Submit your answers to view the result and download the PDF for revision.
Understanding Cyclic Quadrilaterals and Concyclic Points
In Class IX Geometry, a key topic is the study of circles and quadrilaterals inscribed within them. A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. The points lying on the same circle are referred to as concyclic points.
Key Properties
- Sum of Opposite Angles: The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees. Conversely, if the sum of a pair of opposite angles of a quadrilateral is 180 degrees, the quadrilateral is cyclic.
- Angles in the Same Segment: Angles subtended by an arc (or a chord) at any point in the remaining part of the circle are equal.
- Exterior Angle Property: If one side of a cyclic quadrilateral is produced, the exterior angle so formed is equal to the interior opposite angle.
Test for Concyclic Points (Motivation)
There is a specific condition to check if four points lie on a circle. The theorem states: If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the segment, the four points lie on a circle (i.e., they are concyclic).
For example, if A and B are two points, and points C and D are on the same side of line AB such that Angle ACB = Angle ADB, then A, B, C, and D are concyclic.
Quick Revision Table
| Property | Cyclic Quadrilateral | Non-Cyclic (General) |
|---|---|---|
| Vertices | All 4 on a circle | Not all on one circle |
| Opposite Angles | Sum is 180 degrees | Sum is not necessarily 180 degrees |
| Exterior Angle | Equals interior opposite angle | No specific relation |
Practice Questions
- If ABCD is a cyclic quadrilateral and Angle A is 70 degrees, what is the measure of Angle C?
- In a cyclic quadrilateral PQRS, if Angle P – Angle R = 20 degrees, find Angle P.
- Prove that a cyclic parallelogram is a rectangle.
- If two opposite sides of a cyclic quadrilateral are equal, prove that the other two sides are parallel.
- Points A, B, C, D are concyclic. If AC bisects Angle A, prove that AB = AD is not necessarily true unless specific arc conditions are met.
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