Understanding Engineering Mathematics

• A tangent to a circle is perpendicular to the radius drawn through
  the point of contact.

O

Figure 5.23Tangent perpendicular to radius.

One can see this by constructing two radii extended to any two points either side and
equidistant from the point of contact (Figure 5.23). By symmetry, the resulting triangles
have corresponding sides equal and are therefore congruent (152

➤
) and must have corre-
sponding angles equal. This implies that the angle of intersection between the tangent and
radius at point of contact is^12 × 180 °= 90 °.
Two other important results are

• The two tangents drawn from an external point to a circle are equal
  in length.


• The angle between a tangent and a chord through the point of
  contact is equal to the angle subtended by the chord in the alternate
  segment (Figure 5.24).

D

C

B

A

E

∠DAC = ∠ABC

∠EAB = ∠ACB

Figure 5.24Angle in alternate segment.