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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 point B.
Solution
Given: AB + BC + AC = 44 cm
AC = 12 cm
To find: l(BP) , l(BR)
Solution:
`{:("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 ......(iiii)
∴ l(BP) = l(BR) = 10 cm ......[From (i) and (iii)]
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