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.

Answer 1

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)]