A circle is inscribed in a triangle whose sides are 40 cm, 40cm,48cm resp A smaller circle touching two - Maths - Circles

Question

A circle is inscribed in a triangle whose sides are 40 cm,
40cm,48cm resp A smaller circle touching two equal sides of the
triangle and to the first circle Find the radius of smaller circle

Ishwarmani · Accepted Answer

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Student,
Please find
below the solution to the asked query:

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In ∆ABC,AB=AC=40 cm and BC=48 cmWe know that, AD will be median, perpendicular bisector, angle bisector and altitude of the isosceles ∆ABC
So, BD=482=24 cmIn ∆ABD,Using pythagoras theorem,AD2=AB2-BD2⇒AD2=402-242⇒AD2=1600-576=1024⇒AD=1024∴ AD=32 cmAs, centroid divides the median in the ratio of 2:1So, OD=13AD⇒OD=13×32∴ OD=323 cmAlso, AO=23AD⇒AO=23×32∴ AO=643 cmNow, in ∆APE and ∆AOF∠PAE=∠OEF       Common angle∠AEP=∠AFO=90°     Tangent is perpendicular to the radius at the point of contactSo, by AA similarity criteria,∆APE ~ ∆AOF⇒PEOF=APAO⇒r323=AP643⇒AP=2r⇒AO-PO=2r        AO=AP+PO⇒643-r+323=2r⇒643-r-323=2r⇒643-323=2r+r⇒3r=323∴ r=329 cm

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A circle is inscribed in a triangle whose sides are 40 cm, 40cm,48cm resp. A smaller circle touching two equal sides of the triangle and to the first circle. Find the radius of smaller circle.