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.
We introduce the coefficient of proportionality k, then the sides of the quadrilateral are equal to k, 3k, 9k. The fourth side is denoted by x.
By the condition of the problem, the quadrilateral is described around a circle. By the theorem on the described convex quadrilateral, we write the equality of the sums of opposite sides.
k + 9k = 3k + x
x = 7k
The perimeter of the quadrilateral is known by the condition, we get the equation:
k + 3k + 9k + 7k = 20
20k = 20
k = 1
The large side is equal to:
9 * 1 = 9.
Answer: the large side of the quadrilateral is 9 units.
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