A chord AB with a length of 16 cm is drawn in a circle with a radius of 10 cm. The tangents to the circle drawn through points A and B intersect at point C. Find the length of the line segment AC.
The height OH of the isosceles triangle AOB is so its median, then AH = BH = AB / 2 = 16/2 = 8 cm.
In the right-angled triangle AOH, we define the sine of the angle AOH. SinAOH = AH / AO = 8/10 = 0.8.
Since the radius of AO, by the property of the tangent, is perpendicular to the tangent AC, then the triangle AOC is rectangular, with AO = 10 cm, SinAOC = 0.8.
Then CosAOC = √ (1 – 0.64) = √0.36 = 0.6.
OC = AO / CosAOC = 10 / 0.6 = 50/3.
In a rectangular triangle AOC, AC ^ 2 = OC ^ 2 – AO ^ 2 = 2500/9 – 100 = (2500 – 900) / 9 = 1600/9.
AC = 40/3 = 13 (1/3) cm.
Answer: The length of the AC segment is 13 (1/3) cm.
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