Question

find the perimeter of the polygon if ( mangle a=mangle c ).
all the segments are tangent to circle ( o ).

Explanation:

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

If two tangent segments are drawn from an external point to a circle, then the two tangent segments are congruent.
Let the lengths of the tangent segments from \(A\), \(B\), \(C\), and \(D\) to the points of tangency be \(x\), \(y\), \(z\), and \(w\) respectively.
We know that for a quadrilateral circumscribed about a circle (a tangential quadrilateral), the sum of the lengths of two opposite sides is equal to the sum of the lengths of the other two opposite sides.
Let \(AB = 4\) ft, \(BC=10\) ft, \(CD = 8\) ft.
By the property of tangential quadrilaterals \(AB + CD=AD + BC\)

Step2: Calculate the perimeter

The perimeter \(P\) of a quadrilateral \(P=AB + BC+CD + DA\)
Since \(AB + CD=AD + BC\), then \(P = 2(AB + CD)\)
Substitute \(AB = 4\) ft and \(CD = 8\) ft into the formula.
\(P=2(4 + 10)\)
\(P=2\times14\)

Answer:

\(24\)