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.
Answer:
9.4 cm9.4 cm
- 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. - 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. - The tangents from an external point are equal in length. ⟹PA=PB⟹PA=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]
- As corresponding parts of congruent triangles are equal(CPCT), we have ∠PMA=∠PMB and AM=BM Also, [Math Processing Error]
- Now, we can say that OP is the right bisector of AB.
Thus, OP⊥AB and OP bisects AB at M.
Therefore, AM=BM=132 cm=6.5 cm. - Now,
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In right △AMO, we have [Math Processing Error] Therefore, using pythagoras theorem, we have [Math Processing Error] - Also, △APM is a right angled triangle.
Using pythagoras theorem, we have [Math Processing Error] - In right △PAO, using pythagoras theorem [Math Processing Error]
- Now, substituting the value of y in (i), we get
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Thus, PA=9.4 cm.