ABCDEis a circle O . The diameter AC is extended to the point F so that CF=16cm. The line BF is the tangent to the circle at B and FDE is a straight line such that FD=18cm andDE=14cm . The radius of the circle is rcm . Calculate(a) the length in cm of FB(b) the value of r .
Question
ABCDEis a circle O . The diameter AC is extended to the point F so that CF=16cm. The line BF is the tangent to the circle at B and FDE is a straight line such that FD=18cm andDE=14cm . The radius of the circle is rcm . Calculate(a) the length in cm of FB(b) the value of r .
Solution
To solve this problem, we will use the properties of circles and triangles, specifically the Pythagorean theorem and the property that the tangent to a circle is perpendicular to the radius at the point of tangency.
(a) To find the length of FB, we first note that triangle ABC is a right triangle because AB is a radius and BC is a tangent to the circle at B. Therefore, by the Pythagorean theorem, we have AC^2 = AB^2 + BC^2. We also know that AC = AB + BC = AB + 16 cm.
We can set up the equation as follows:
(AB + 16)^2 = AB^2 + 16^2 AB^2 + 32AB + 256 = AB^2 + 256 32AB = 256 AB = 8 cm
Then, we can find FB by adding AB and BF:
FB = AB + BF = 8 cm + 16 cm = 24 cm
(b) To find the radius r of the circle, we note that the radius is equal to AB, which we found to be 8 cm. Therefore, r = 8 cm.
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