The incircle of Delta ABC touches the sides o | vedclass

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Question

(A) Let $AP = AR = x$, $BP = BQ = y$, and $CQ = CR = z$ (tangents from an external point to a circle are equal in length).Given:$AB = x + y = 14$ $(1)

Vedclass · Accepted Answer

$AP = 5, BQ = 9, RC = 2$

Vedclass · Answer

(A) Let $AP = AR = x$, $BP = BQ = y$, and $CQ = CR = z$ (tangents from an external point to a circle are equal in length).Given:$AB = x + y = 14$ $(1)$$BC = y + z = 11$ $(2)$$CA = z + x = 7$ $(3)$Adding equations $(1)$, $(2)$, and $(3)$:$2(x + y + z) = 14 + 11 + 7 = 32$$x + y + z = 16$Now, subtracting each equation from the sum:$z = (x + y + z) - (x + y) = 16 - 14 = 2$$x = (x + y + z) - (y + z) = 16 - 11 = 5$$y = (x + y + z) - (z + x) = 16 - 7 = 9$Thus, $AP = x = 5$, $BQ = y = 9$, and $RC = z = 2$.

The incircle of $\Delta ABC$ touches the sides $\overline{AB}$, $\overline{BC}$ and $\overline{CA}$ at points $P$, $Q$ and $R$ respectively. If $AB = 14$, $BC = 11$ and $CA = 7$, find the lengths of $AP$, $BQ$ and $RC$.

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