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Tangent Properties



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Mathematics    Geometry    Tangent Properties

Tangent Properties:

    ? A tangent is perpendicular to the radius at the point of contact.
    ? A tangent is perpendicular to the radius drawn to the point of contact.
    ? A line tangent to a circle if it is perpendicular to a radius at its outer end.
    ? A line passes through the center of a circle if it is perpendicular to a tangent at its point of
      contact.
    ? Tangents to a circle from an outside point are congruent. 
    ? The segment from the center of a circle to an outside point bisects the angle between then
      tangents from the point to the circle.

Example: Find the value of the pro-numeral(s) in each of the following diagrams:




Solution:

a.   OPT = 900 ( as OP?TP)
      In OPT,
      x + 56 + 90 = 180 (Sum of angles of triangle is 180)
      x + 146 = 180
      or x = 34

b.  OPA = 900 ( as OP?PA)
     By Pythagoras Theorem in OPA,
     x2 + 52 = 132
     x2 + 25 = 169
     x2 = 144
     x = = 12

c  .x = 75 (as PO bisects QOR)
     In OPQ,
     a + 75 + 90 = 180 (Sum of angles of triangle is 180)
     a + 165 = 180
     or a = 15
     b = a = 15 (as PO bisects QOR)

Angle-Measurement Principles:

    ? A central angle is measured by its intercepted arc.
    ? An inscribed angle is measured by one-half its intercepted arc.
    ? In the same or congruent circles, congruent inscribed angles have congruent intercepted arcs.
    ? In the same or congruent circles, inscribed angles having congruent intercepted arcs are
      congruent.
    ? Angles inscribed in the same or congruent arcs are congruent.
    ? An angle inscribed in a semicircle is a right angle.



    ? Opposite angles of an inscribed quadrilateral are supplementary.
    ? Parallel lines intercept congruent arcs on a circle.
    ? An angle formed by a tangent and a chord is measured by one-half its intercepted arc.
    ? An angle formed by two intersecting chords is measured by one-half the sum of the intercepted
      arcs.
    ? An angle formed by two secants intersecting outsides a circle is measured by one-half the
      difference of the intercepted arcs.
    ? An angle formed by a tangent and a secant intersecting outsides a circle is measured by one-
      half the difference of the intercepted arcs.
    ? An angle formed by two tangents intersecting outside a circle is measured by one-half the
      difference of the intercepted arcs.



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