Understanding Circle Properties: Identifying the Incorrect Statement

The question asks us to identify the statement that is INCORRECT among the given options regarding the properties of a circle. Let's examine each statement carefully based on standard geometric principles related to circles.

Analyzing Each Statement on Circle Properties

We will evaluate each statement to determine its validity.

1. Statement 1: Two tangents drawn at the end of the diameter of a circle are parallel.

   Let's consider a circle with centre O and a diameter AB. Let tangents be drawn at points A and B. The radius at the point of contact is perpendicular to the tangent. So, OA is perpendicular to the tangent at A, and OB is perpendicular to the tangent at B. Since A, O, and B are collinear (AB is a diameter), OA and OB lie on the same line which is perpendicular to both tangents. Lines perpendicular to the same line are parallel to each other. Therefore, the tangents drawn at the ends of a diameter are indeed parallel.

   This statement is CORRECT.

2. Statement 2: The radius drawn perpendicular to a chord bisects the chord.

   This is a fundamental property of circles. If a radius (or part of a radius) is drawn from the center of a circle perpendicular to a chord, it divides the chord into two equal segments. Conversely, a line segment from the center that bisects a chord is perpendicular to the chord.

   This statement is CORRECT.

3. Statement 3: The diameter of a circle is the longest chord of a circle.

   A chord is a line segment connecting two points on the circle. The diameter is a special type of chord that passes through the center of the circle. Any chord that does not pass through the center will have a length less than the diameter. The chord passing through the center has the maximum possible length. Therefore, the diameter is the longest chord.

   This statement is CORRECT.

4. Statement 4: The perpendicular distance from the centre of a circle increases when the length of a chord increases.

   Let's consider a chord in a circle. The perpendicular distance from the center to the chord can be related to the length of the chord using the Pythagorean theorem. If 'r' is the radius, 'd' is the perpendicular distance from the center to the chord, and 'l' is half the length of the chord (\(l = \frac{\text{chord length}}{2}\)), then in the right-angled triangle formed by the radius, the distance 'd', and half the chord, we have \(r^2 = d^2 + l^2\). This can be rewritten as \(d^2 = r^2 - l^2\).

   Since 'r' (the radius) is constant for a given circle, as the length of the chord increases, 'l' (half the chord length) also increases. From the equation \(d^2 = r^2 - l^2\), if \(l\) increases, \(l^2\) increases. Since \(r^2\) is constant, \(r^2 - l^2\) must decrease for \(l^2\) to increase. Therefore, \(d^2\) decreases, which means 'd' (the perpendicular distance) decreases.

   Conversely, as the length of the chord decreases (getting closer to a point, or length 0), the distance 'd' increases (approaching the radius 'r' for a point, or 'r' for a tangent). The longest chord (the diameter) has a length of \(2r\), which means \(l = r\). In this case, \(d^2 = r^2 - r^2 = 0\), so \(d = 0\), meaning the distance from the center is 0, as the diameter passes through the center. This shows that the maximum chord length corresponds to the minimum distance (0) from the center.

   So, the perpendicular distance from the center of a circle decreases when the length of a chord increases.

   This statement is INCORRECT.

Based on our analysis, the INCORRECT statement with respect to the properties of a circle is the fourth statement.

Here's a summary table showing the relationship between chord length and distance from the center:

Identifying the Incorrect Statement

The statement that claims "The perpendicular distance from the centre of a circle increases when the length of a chord increases" is factually wrong. The distance from the center to a chord actually decreases as the chord length increases.

Revision Table: Key Circle Properties

Additional Information: Understanding Circle Geometry

Circle geometry deals with the properties of circles and their associated lines and segments like radii, diameters, chords, secants, and tangents. These properties are fundamental in geometry and have numerous applications.

• A radius is a line segment from the center to any point on the circle. All radii of the same circle are equal in length.
• A diameter is a chord passing through the center. It is twice the length of the radius (\(D = 2r\)).
• A chord is a line segment connecting two points on the circle.
• A secant is a line that intersects the circle at two points.
• A tangent is a line that touches the circle at exactly one point, called the point of contact. The radius at the point of contact is perpendicular to the tangent.
• Congruent chords in a circle are equidistant from the center. Conversely, chords equidistant from the center are congruent.

Mastering these basic properties is crucial for solving problems related to circles in geometry.