PA AND PB ARE TANGENTS SUCH THAT PA = 9 CM AND ANGLE APB =60 FIND THE LENGTH OF - Maths - Circles

Question

PA AND PB ARE TANGENTS SUCH THAT PA = 9 CM AND ANGLE APB =60
FIND THE LENGTH OF THE CHORD AB

Prashasti Singh Dahiya · Accepted Answer

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Given : PA and PB are tangents of a circle, PA = 9 cm and ∠APB = 60° Let O be the center of the given circle and C be the point of intersection of OP and AB In ΔPAC and ΔPBC PA = PB ( $n n âˆµ n n$ Tangents from an external point are equal) ∠APC = ∠BPC ( $n n âˆµ n n$ Tangents from an external point are equally inclined to the segment joining center to that point) PC = PC ( $n n âˆµ n n$ Common) Thus ΔPAC $n n ≅ n n$ ΔPBC (By SAS congruency rule) (1) ∴ AC = BC Also ∠APB = ∠APC + ∠BPC $n n \Rightarrow n ∠ n APC = n \frac{n}{n 1 n} n ∠ n APB ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½(n ∠ n APC = n ∠ n BPC n) n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n = n \frac{n}{n 1 n} n ï¿½ n â¢ n n n 60 n ï¿½ n â¢ n = n n 30 n ï¿½ n â¢ n n$ ∠ACP + ∠BCP = 180° $n n \Rightarrow n ∠ n ACP = n \frac{n}{n 1 n} n ï¿½ n â¢ n n n 180 n ï¿½ n â¢ n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n (n ∠ n ACP = n ∠ n BCP n) n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n = n n 90 n ï¿½ n n$ Now in right triangle ACP sin 30 = AC/PA = 1/2 =AC/9 AC = 4 5 cm ∴ AB = AC + BC = AC + AC ( $n n âˆµ n n$ AC = BC) ⇒ AB = (4 5 +4 5 ) cm = 9 cm

PA AND PB ARE TANGENTS SUCH THAT PA = 9 CM AND ANGLE APB =60. FIND THE LENGTH OF THE CHORD AB