Inscribed angles and tangents
An inscribed angle has its vertex on the circle itself. Its measure is always half the arc it cuts off, a fact with remarkable consequences.
Inscribed angles
An inscribed angle has its vertex on the circle and its sides as chords. The arc lying inside the angle is its intercepted arc. The inscribed angle theorem says an inscribed angle measures half its intercepted arc: an arc of 80 degrees gives an inscribed angle of 40 degrees.
Same arc, same angle
All inscribed angles that intercept the same arc are congruent. This explains why, in a theatre with a curved front row, seats along a circle through the screen's edges see the screen at the same angle.
Angles in a semicircle
An inscribed angle that intercepts a semicircle is a right angle, since half of 180 degrees is 90. Thales of Miletus is credited with this theorem. Carpenters use it: placing the corner of a set square on a circular object's edge and marking where the sides cross gives a diameter.
Cyclic quadrilaterals
A quadrilateral with all four vertices on a circle is cyclic. Its opposite angles are supplementary, since each pair intercepts arcs totalling the whole 360-degree circle. If one angle of a cyclic quadrilateral is 105 degrees, the opposite angle is 75 degrees.
Tangent lines
A tangent is perpendicular to the radius at the point of tangency. This lets us use right triangles: if a tangent from point P touches a circle of radius 5 at T, and P is 13 from the centre, then the tangent segment PT is the square root of (169 - 25) = 12.
Tangent segments from a point
The two tangent segments drawn to a circle from the same outside point are congruent. This is why a circle inscribed in a triangle touches each side at points that split the sides into matching pairs of lengths.
Angles formed outside or inside
Two chords crossing inside a circle form an angle equal to half the sum of the intercepted arcs. Two secants or tangents meeting outside form an angle equal to half the difference of the intercepted arcs. With arcs of 100 and 40 degrees, the inside angle is 70 and the outside angle 30.
Worked example
From a lighthouse 120 m above sea level, the line of sight to the horizon is tangent to Earth, perpendicular to Earth's radius, about 6,371 km. The tangent distance is the square root of ((6,371,000 + 120) squared - 6,371,000 squared), which works out to about 39,100 m: the horizon is roughly 39 km away.
Key takeaways
An inscribed angle measures half its intercepted arc, so inscribed angles on the same arc are congruent and an angle inscribed in a semicircle is a right angle (Thales). Opposite angles of a cyclic quadrilateral are supplementary. A tangent is perpendicular to the radius at the point of contact, creating right triangles. Tangent segments from one outside point are congruent. Chords crossing inside make angles of half the sum of the arcs; secants or tangents meeting outside make angles of half the difference. Tangents to Earth give the distance to the horizon.
Flashcard
Vertex on the circle
Check your understanding
An inscribed angle intercepts an arc of 80 degrees. What does the angle measure?