# Three consecutive sides of a quadrilateral circumscribed about a circle

Three consecutive sides of a quadrilateral circumscribed about a circle are related as 3: 4: 5. the perimeter of this quadrangle is 48 cm.Find the lengths of its sides

Since a circle is inscribed in a quadrilateral, the sums of the lengths of the opposite sides of such a quadrilateral are equal.

AB + CD = BC + AD.

Let the length of the segment AB = 3 * X cm, then BC = 4 * X cm, CD = 5 * X cm.

AB + CD = 3 * X + 5 * X = 8 * X cm.

Then BC + AD = 8 * X = 4 * X + AD.

AD = 8 * X – 4 * X = 4 * X cm.

The perimeter of the quadrangle will be: P = 2 * (AB + CD) = 2 * 8 * X = 16.

X = 48/16 = 3.

AB = 3 * 3 = 9 cm, BC = 4 * 3 = 12 cm, CD = 5 * 3 = 15 cm, AD = 4 * 3 = 12 cm.

Answer: The sides of the quadrangle are equal: 9 cm, 12 cm, 15 cm, 12 cm.

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