7.5 Some properties of tangents, secants and chords
Theorem: If the tangent to a circle and the radius of the circle intersect they do so at right angles :
7.9 (a) Figure
In figure 7.9 (a) l
is a tangent to the circle at A and PA is the radius.
To prove that PA is perpendicular to l
, assume that it is not.
Now, with reference to figure 7.9 (b) drop
a perpendicular from P onto
l at say B. Let D be a point
on l such that
B is the midpoint of AD.
In figure 7.9 (b) consider D
PDB and D PAB
seg.BA ( B is the midpoint of AD)
( PB is perpendicular to l
seg.PB = seg.PB (same segment)
D PAB (SAS)
seg.PD @ seg.PA corresponding
sides of congruent triangles are congruent.
D is definitely a point on the circle because l
(seg.PD) = radius.
D is also on
l which is the tangent. Thus
l intersects the
circle at two distinct points A and D. This contradicts the definition
of a tangent.
Hence the assumption that PA is not perpendicular
is false. Therefore PA is perpendicular to l.
Angles formed by intersecting chords, tangent and chord and two secants: If two chords intersect in a circle, the angle they form is half the sum of the intercepted arcs.
In the figure 7.10 two chords AB and CD intersect
at E to form Ð1
and Ð 2.
m Ð1 = (m
seg.AD + m seg.BC) and
m Ð2 = (m
seg.BD + m seg.AC)
Tangent Secant Theorem: If a chord intersects
the tangent at the point of tangency, the angle it forms is half
the measure of the intercepted arc. In the figure 7.11 l
is tangent to the circle. Seg.AB which is a chord, intersects it
at B which is the point of tangency.
The angles formed Ð
ABX and Ð ABY are half
the measures of the arcs they intercept.
m Ð 1 = m
m Ð 2 = m
This can be proved by considering the three following cases.
O is the center of the circle
Line DBC is tangent to it at B.
BA is the chord in question. X is a point on the circle on the C side of BA and Y is on the D side by BA.