Consider the following statements : 1. If two chords AB and AC of a circle are equal, then the centre of the circle lies on the angle bisector of angle CAB. 2. If two concentric circles are intersected by a line at A, B, C and D respectively, then AC = BD. Which of the above statements is/are correct?
Both 1 and 2
Analyzing Circle Properties and Statements
The question asks us to evaluate two statements related to the properties of circles, chords, angle bisectors, and concentric circles. We need to determine if each statement is correct.
Statement 1: Equal Chords and Angle Bisector
The first statement says: If two chords AB and AC of a circle are equal, then the centre of the circle lies on the angle bisector of angle CAB.
Let O be the center of the circle. We are given that chord AB is equal to chord AC, i.e., AB = AC.
Consider the triangles formed by the center O and the endpoints of the chords: ▵OAB and ▵OAC. In these two triangles:
- OA is a common side to both triangles.
- OB = OC because they are both radii of the same circle.
- AB = AC (Given in the statement).
By the SSS (Side-Side-Side) congruence criterion, ▵OAB is congruent to ▵OAC.
Since the triangles are congruent, their corresponding angles are equal. Therefore, the angle ∠OAB is equal to the angle ∠OAC. This means that the line segment OA bisects the angle ∠CAB.
Since OA is the line segment joining the center O to the vertex A of the angle ∠CAB, and it bisects the angle, the center O lies on the angle bisector of ∠CAB.
Thus, Statement 1 is correct.
Statement 2: Concentric Circles and Line Intersection
The second statement says: If two concentric circles are intersected by a line at A, B, C and D respectively, then AC = BD.
Let O be the center of the two concentric circles. Let a line intersect the outer circle at points A and D, and the inner circle at points B and C. Assume the points are in the order A, B, C, D along the line.
Draw a perpendicular from the center O to the line. Let the foot of the perpendicular be M.
A fundamental property of circles is that the perpendicular from the center to a chord bisects the chord.
- For the outer circle, AD is a chord. The perpendicular OM bisects AD. So, AM = MD.
- For the inner circle, BC is a chord. The perpendicular OM bisects BC. So, BM = MC.
Now consider the segments AC and BD based on the order A, B, C, D on the line:
- AC = AM - CM
- BD = DM - BM
Since AM = DM and CM = BM, we can substitute these equalities:
- AC = AM - BM
- BD = AM - BM
Therefore, AC = BD.
This holds true regardless of the order of B and C relative to A and D, as long as A and D are on one circle and B and C are on the other concentric circle, and the line passes through both.
Thus, Statement 2 is correct.
Conclusion on Statements
Based on our analysis, both Statement 1 and Statement 2 are correct.
| Statement | Analysis | Correctness |
|---|---|---|
| 1. Equal chords AB=AC implies center on angle bisector of ∠CAB. | Using SSS congruence for ▵OAB and ▵OAC proves ∠OAB = ∠OAC. | Correct |
| 2. Line intersects concentric circles at A, B, C, D implies AC = BD. | Using perpendicular from center bisects chord (AM=MD, BM=MC), leads to AC = AM-CM and BD = DM-BM, which implies AC = BD. | Correct |
Revision Table: Key Concepts in Circle Geometry
Reviewing the concepts used:
- Chord: A line segment connecting two points on a circle.
- Angle Bisector: A line or ray that divides an angle into two equal angles.
- Concentric Circles: Two or more circles that share the same center point.
- Perpendicular from Center to Chord: A line segment drawn from the center of a circle perpendicular to a chord; it always bisects the chord.
- Congruence of Triangles (SSS Rule): If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.
Additional Information: Further Circle Properties
Here are some more important properties related to circles and chords:
- Equal chords of a circle are equidistant from the center. Conversely, chords equidistant from the center are equal.
- The angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the remaining part of the circle.
- Angles in the same segment of a circle are equal.
- The angle in a semicircle is a right angle (90 degrees).
- A tangent at any point of a circle is perpendicular to the radius through the point of contact.
Understanding these fundamental theorems is crucial for solving problems in circle geometry.
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