Question

1. find the perimeter of \\( \triangle def \\).
2. find the perimeter of

quadrilateral \\( pqrs \\).

Explanation:

Step1: Use the property of tangents from a common external point

Tangents from a common external point to a circle are equal.
Let \(DG = DH = x\), \(EH=EI = y\), \(FI = FG = z\).
We know \(DG = 11\) (since \(DG = DH\)), \(EH = 16\) (since \(EH=EI\)), \(FI=9.5\) (since \(FI = FG\)).

Step2: Calculate the perimeter of \(\triangle DEF\)

The perimeter of \(\triangle DEF\) is \(DE + EF+FD\).
\(DE=DH + HE\), \(EF = EI+FI\), \(FD=FG + GD\).
The perimeter \(P=(DH + HE)+(EI + FI)+(FG + GD)\).
Since \(DG = DH\), \(EH=EI\), \(FI = FG\), the perimeter \(P = 2(DG + EH+FI)\).
Substitute \(DG = 11\), \(EH = 16\), \(FI = 9.5\) into the formula.
\(P=2(11 + 16+9.5)\)
\(P=2\times36.5\)
\(P = 73\)

Answer:

The perimeter of \(\triangle DEF\) is \(73\)