In the figure, radius of the circle with centre at $O$ is $6$ centimetres. And $P A = 4 c m, P B = 5 c m$ . Find the length of $O P$ .

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Solution

From $O$ , draw $O D ⊥ A B$
$A B = A P + P B = 4 + 5 = 9 c m$
$\Rightarrow$ $A D = 4.5 c m$
In right $△ O A D$ ,
$O D = \sqrt{{O A}^{2} - {A D}^{2}} = \sqrt{6^{2} - {4.5}^{2}} = \sqrt{36 - 20.25} = \sqrt{15.75}$
$P D = A D - A P = 4.5 - 4 = 0.5 c m$
In right $△ O P D$ ,
$O P = \sqrt{{O D}^{2} + {P D}^{2}} = \sqrt{15.75 + {0.5}^{2}} = \sqrt{15.75 + 0.25} = \sqrt{16} = 4 c m$

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In the figure, radius of the circle with centre at O is 6 centimetres. And PA=4cm,PB=5cm. Find the length of OP.

From O, draw OD⊥ABAB=AP+PB=4+5=9cm⇒ AD=4.5cmIn right △OAD,OD=√OA2−AD2=√62−4.52=√36−20.25=√15.75PD=AD−AP=4.5−4=0.5cmIn right △OPD,OP=√OD2+PD2=√15.75+0.52=√15.75+0.25=√16=4cm

In the figure, radius of the circle with centre at $O$ is $6$ centimetres. And $P A = 4 c m, P B = 5 c m$ . Find the length of $O P$ .