In the given figure, ABAB is a chord of length 13 cm13 cm of a circle with center OO and radius 9 cm9 cm. The tangents at AA and BB intersect at PP. Find the length of PAPA.

O B A P


Answer:

9.4 cm9.4 cm

Step by Step Explanation:
  1. Given:
    Length of chord ABAB is 13 cm13 cm.
    The radius of the circle is 9cm9cm.
    The tangents at AA and BB intersect at PP.
  2. Here, we have to find the length of PAPA.

    Now, join OO to PP such that OPOP intersects ABAB at MM.

    Let PA=x cmPA=x cm and PM=y cmPM=y cm.
    x cm 13 cm 9 cm O B A P y cm M
  3. The tangents from an external point are equal in length. PA=PBPA=PB Also, two tangents to a circle from an external point are equally inclined to the line segment joining the center to that point. [Math Processing Error] Also, [Math Processing Error] By SAS Congruence Criterion, we conclude [Math Processing Error]
  4. As corresponding parts of congruent triangles are equal(CPCT), we have PMA=PMB and AM=BM Also, [Math Processing Error]
  5. Now, we can say that OP is the right bisector of AB.

    Thus, OPAB and OP bisects AB at M.

    Therefore, AM=BM=132 cm=6.5 cm.
  6. Now, [Math Processing Error]
    x cm 9 cm 6.5 cm 6.5 cm O B A P y cm M

    In right AMO, we have [Math Processing Error] Therefore, using pythagoras theorem, we have [Math Processing Error]
  7. Also, APM is a right angled triangle.
    Using pythagoras theorem, we have [Math Processing Error]
  8. In right PAO, using pythagoras theorem [Math Processing Error]
  9. Now, substituting the value of y in (i), we get [Math Processing Error]
    Thus, PA=9.4 cm.

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