Through the ends of the chord AB of the circle with center O, tangents are drawn, which intersect at point C.

| February 15, 2021 | education

Through the ends of the chord AB of the circle with center O, tangents are drawn, which intersect at point C. Find the degree measure of the smaller of the arcs AB if AO = 6 cm and the perimeter of the quadrilateral AOBC is 24 cm.

Draw the radius OB from the center O circle.

By condition, ОА = 6 cm, then ОВ = 6 cm.

By the property of tangents, the angle OAS = OBC = 90, and the segment AC = BC.

Since the perimeter of the quadrangle ОАСВ = 24 cm, then (АС + ВС) = 24 – 6 – 6 = 12 cm.

AC = BC = 12/2 = 6 cm.

In a quadrangle OACB, all sides are equal and two opposite angles are equal to 90, hence the quadrilateral is a square.

Then the central angle AOB = 90, and hence the degree measure of the smaller arc AB = 9.

Answer: The degree measure of the smaller arc AB is 90.



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