The three sides of a quadrilateral circumscribed about a circle

The three sides of a quadrilateral circumscribed about a circle are in sequential order as 1: 3: 9 find the largest side of this quadrilateral if you know that its perimeter is 20

Let ABCD be a circumscribed quadrangle (a circle is inscribed in it): AB: BC: CD = 1: 3: 9.
Let x be the coefficient of proportionality, then: AB = x, BC = 3x, CD = 9x.
The perimeter of ABCD is:
P = AB + BC + CD + AD;
x + 3x + 9x + AD = 20;
AD = 20 – 13x.
It is known that a circle can be inscribed into a quadrilateral only when the sums of its opposite sides are equal, that is:
AB + CD = AD + BC.
Then:
x + 9x = 20 – 13x + 3x;
10x = 20 – 10x;
20x = 20;
x = 20/20;
x = 1.
Then: AB = x = 1, BC = 3x = 3 * 1 = 3, CD = 9x = 9 * 1 = 9, AD = 20 – 13x = 20 – 13 * 1 = 20 – 13 = 7.
Answer: large side of CD = 9.




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