#### Question

AB is a line segment and M is its mid – point three semi circles are drawn with AM, MB and AB as diameter on the same side of the line AB. A circle with radius r unit is drawn so that it touches all the three semi circles show that : AB = 6 × r

#### Solution

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Let O, P and Q be the centers of the circle and semicircles.

Join OP and OQ.

OR = OS = r

and AP = PM = MQ = QB =`(AB)/4`

Now, OP = OR + RP = r + `(AB)/4` (since PM=RP=radii of same circle)

Similarly, OQ = OS + SQ = r + `(AB)/4`

OM = LM -; OL =`(AB)/2`-r

Now in Rt. ΔOPM,

`OP^2 = PM^2 + OM^2`

⇒ `(r = (AB)/4)^2 = ((AB)/4)^2 +((AB)/2 - r)^2`

⇒ `r^2 + "AB"^2/16 + "rAB"/2 = "AB"^2/16 + "AB"^2/4 + r^2 - rAB`

⇒` "rAB"/2 = "AB"^2/4 - rAB`

⇒` "AB"^2/4 = "rAB"/2 + rAB`

⇒ ` "AB"^2/4 = "3rAB"/2`

⇒ `"AB"/4 = 3/2 r`

⇒ `AB = 3/2 rxx 4 = 6 r`

Hence AB = 6 × r

Is there an error in this question or solution?

Solution Ab is a Line Segment and M is Its Mid – Point Three Semi Circles Are Drawn with Am, Mb and Ab as Diameter on Same Side of Line Ab.A Circle with Radius R Unit is Drawn So Three Semi Circles Ab = 6 × R Concept: Arc and Chord Properties - Angle in a Semi-circle is a Right Angle.