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Question
The figure ΔABC is an isosceles triangle with a perimeter of 44 cm. The sides AB and BC are congruent and the length of the base AC is 12 cm. If a circle touches all three sides as shown in the figure, then find the length of the tangent segment drawn to the circle from the point B.
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Solution
Given: AB + BC + AC = 44 cm
AC = 12 cm
To find: `l`(BP), `l`(BR)
`{:(seg AP ≅ seg AQ),(seg QC ≅ seg RC),(seg BP ≅ seg BR):}}` ...[Tangent Segment theorem]
`{:("Let" l("AP") = l("AQ") = x","),(l("QC") = l("RC") = y","),(l("BP") = l("BR") = "z"):}}` ...(i)
AC = AQ + QC ...[A – Q – C]
∴ AC = x + y
∴ x + y = 12 ...(ii) [Given]
AB + BC + AC = 44 ...[Given]
∴ (AP + PB) + (BR + RC) + (AQ + QC) = 44 ...[A–P–B, B–R–C, A–Q–C]
∴ x + z + z + y + x + y = 44 ...[From (i)]
∴ 2x + 2y + 2z = 44
∴ 2(x + y) + 2z = 44
∴ 2(12) + 2z = 44 ...[From (ii)]
∴ 24 + 2z = 44
∴ 2z = 44 – 24
∴ 2z = 20
∴ z = 10 ...(iii)
∴ `l`(BP) = `l`(BR) = 10 cm ...[From (i) and (iii)]
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