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Learn the fundamental properties of tangents to circles with clear definitions and theorems. Practice a curated set of geometry problems involving tangents, radii, chords, and intersecting lines. Ideal for students preparing for Olympiad-level maths or anyone looking to deepen their understanding of circle geometry.
Definition. A tangent to a circle is a line that touches the circle at exactly one point.
Theorem 1. If a line passes through a point on a circle and is perpendicular to the radius drawn to that point, then it is a tangent to the circle.
Theorem 2. The radius drawn to the point of tangency is perpendicular to the tangent line.
1. Find the geometric locus of the centers of circles that are tangent to both sides of a given angle.
2. Let AB be the diameter of a circle with center O. Point K lies on the circle.
a) Find ∠KOA if ∠KVA = 26°.
b) Find ∠KVA if ∠KOA = 26°.
3. Two circles are tangent to the same line from opposite sides. The line connecting their centers intersects the tangent line at an angle of 30°.
Find the distance between the centers if the radii of the circles are:
a) 2 cm and 5 cm
b) 3 cm and 6 cm
4. Two lines are tangent to a circle with center O at points A and B, and they intersect at point C.
Find the angle between these lines if:
a) ∠ABO = 15°
b) ∠ABO = 43°
5. Two circles share the same center. Chords AB and AC of the larger circle are tangent to the smaller circle. The angle ∠BAC is 60°.
a) Find the radius of the larger circle if the radius of the smaller circle is 4 cm.
b) Find the radius of the smaller circle if the radius of the larger circle is 6 cm.
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