Question

In \[\vartriangle ABC\], a circle is inscribed. If \[AB = 13\]cm, \[BC = 14\]cm and \[AE = 7\]cm. Find \[AC\].

Answer 1

Hint: We use the property of tangents that tangents from an external point to the circle are equal in length. We divide three sides of the triangle as three sets of tangents from three points and use the given values of sides to find the length of AC.

Complete step-by-step answer:
We are given a \[\vartriangle ABC\] in which circle is inscribed such that \[AB = 13\]cm, \[BC = 14\]cm and \[AE = 7\]cm
Since we know tangents from an external point to a circle are equal in length, we write the set of tangents that are equal in length.
AF and AE are tangents from point A to the circle
\[ \Rightarrow AF = AE\]
BF and BD are tangents from point B to the circle
\[ \Rightarrow BF = BD\]
CD and CE are tangents from point C to the circle
\[ \Rightarrow CD = CE\]
We are given \[AE = 7\]cm
We know \[AF = AE\] as they are tangents from point A to the circle.
\[ \Rightarrow AF = 7\]cm
We know the length \[AB = 13\]cm
We can write \[AB = AF + FB\] … (1)
Substituting the values of AB and AF in equation (1)
\[ \Rightarrow 13 = 7 + FB\]
Shift all constants to one side of the equation.
\[ \Rightarrow FB = (13 - 7)\]cm
\[ \Rightarrow FB = 6\]cm
We know \[FB = BD\] since they are tangents from point B to circle
\[ \Rightarrow BD = 6\]cm
We know the length \[BC = 14\]cm
We can write \[BC = BD + DC\] … (2)
Substituting the values of BC and BD in equation (2)
\[ \Rightarrow 14 = 6 + DC\]
Shift all constants to one side of the equation.
\[ \Rightarrow DC = (14 - 6)\]cm
\[ \Rightarrow DC = 8\]cm
We know \[DC = CE\] since they are tangents from point C to circle
\[ \Rightarrow CE = 8\]cm
We know \[AC = AE + CE\]
Substitute the value of \[AE = 7\]cm and \[CE = 8\]cm
\[ \Rightarrow AC = (7 + 8)\]cm
\[ \Rightarrow AC = 15\]cm

So, the value of AC is 15cm.

Note: Students many times get confused between the pair of tangents that are equal and end up doing wrong calculations. Keep in mind we take the vertices of the triangle as the external points and the tangents are from that point to the circle, always write first the point associated with the equal tangents and then proceed with the solution.