Circles

A circle is a collection of all points in a plane that are at a constant distance from a fixed point. The fixed point is called the center, and the constant distance is known as the radius. For example, the rim of a wheel is a circle with the hub as its center.

A circle consists of several parts including the radius (distance from center to any point on the circle), diameter (twice the radius), chord (a line segment joining two points on the circle), arc (a part of the circumference), sector (region bounded by two radii and an arc), and segment (region bounded by a chord and an arc).

A tangent to a circle is a line that touches the circle at exactly one point, known as the point of contact. It is perpendicular to the radius at the point of contact. For instance, the path of a bicycle wheel is tangent to the ground at the point of contact.

To prove a line is tangent to a circle, show that it intersects the circle at exactly one point and is perpendicular to the radius at that point. This can be done using the Pythagorean theorem or by showing that the distance from the center to the line equals the radius.

A secant is a line that intersects a circle at two distinct points, while a tangent touches the circle at exactly one point. The tangent is perpendicular to the radius at the point of contact, whereas a secant is not.

From a point outside the circle, exactly two tangents can be drawn to the circle. These tangents are equal in length and symmetric with respect to the line joining the point to the center.

The length of a tangent from an external point P to a circle is given by √(OP² - r²), where O is the center, r is the radius, and OP is the distance from P to O. This formula is derived from the Pythagorean theorem.

Consider two tangents PQ and PR from an external point P to a circle with center O. Triangles OQP and ORP are congruent by RHS (right angle, hypotenuse, side), as OQ = OR (radii), OP is common, and both are right-angled. Hence, PQ = PR by CPCT.

The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the center. For example, if the angle at the center is θ, the angle between the tangents is 180° - θ.

If the length of the tangent (l) from a point at distance (d) from the center is known, the radius (r) can be found using the formula r = √(d² - l²). This is derived from the right triangle formed by the radius, tangent, and the line joining the point to the center.

Tangents drawn from an external point to two concentric circles are equal in length. This is because the distance from the external point to the center is the same for both circles, and the difference in radii does not affect the tangent lengths.

To construct a tangent from an external point P, join P to the center O. Find the midpoint M of OP and draw a circle with diameter OP. The intersection points of this circle with the original circle are the points of contact. Lines joining P to these points are the tangents.

The point of contact is where the tangent touches the circle. At this point, the tangent is perpendicular to the radius. This property is crucial in proving theorems related to tangents and in solving geometric problems involving circles.

Yes, a tangent can be parallel to a chord. The tangent at the endpoint of a diameter is parallel to any chord perpendicular to that diameter. This is because both the tangent and the chord are perpendicular to the same radius.

The angle between a tangent and a chord is equal to the angle in the alternate segment. This is known as the alternate segment theorem. For example, if a tangent at A meets chord BC at A, then angle BAC equals the angle in the segment opposite to it.

The distance between two parallel tangents of a circle is equal to the diameter of the circle. This is because each tangent is perpendicular to a radius, and the two radii lie on the same straight line, making the distance between them 2r (diameter).

A circle can have a maximum of two parallel tangents. These are the tangents at the endpoints of a diameter, as they are both perpendicular to the same diameter and hence parallel to each other.

In a quadrilateral circumscribing a circle, the sum of the lengths of opposite sides are equal. This is because the lengths of the two tangents drawn from an external point to a circle are equal. Applying this to all four vertices gives AB + CD = AD + BC.

The angle between the two tangents drawn from an external point to a circle is bisected by the line joining the point to the center. This means the line from the external point to the center divides the angle between the tangents into two equal parts.

The area of a sector of a circle is given by (θ/360) × πr², where θ is the central angle in degrees and r is the radius. For example, a semicircle has θ = 180°, so its area is (180/360) × πr² = (1/2)πr².

The length of an arc of a circle is given by (θ/360) × 2πr, where θ is the central angle in degrees and r is the radius. For instance, the circumference is the arc length when θ = 360°, which is 2πr.

If the length of an arc (l) and the central angle (θ) are known, the radius (r) can be found using the formula r = (l × 360)/(2πθ). This rearranges the arc length formula to solve for the radius.

The angle subtended by an arc at the center is twice the angle subtended at any point on the remaining part of the circle. This is known as the central angle theorem. For example, if the central angle is 60°, the angle at the circumference is 30°.

In a cyclic quadrilateral, the sum of opposite angles is 180°. This is because each angle is half the sum of the arcs intercepted by the opposite sides. Since the total arcs around the circle sum to 360°, the opposite angles must add up to 180°.