CB is tangent to ⊙A at point C. Find the radius.

Circle A is shown. Line segment A C is a radii. Line segment B C is a tangent and it intersects with the circle at point C. A line is drawn from point B to point A and a point is drawn where the line intersects with the circle. The length of the radius is r, the length of C B is 8, and the length of B to the circle is 5.

CB ⊥ AC by the radius-tangent theorem, so ∠C is a right angle.

ΔABC is a right triangle, so apply the Pythagorean theorem.

Use the steps and solve for the radius.

r2 + 82 = (r + 5)2
r2 + 64 = r2 + 10r + 25

In geometry, the radius is defined as a line segment joining the center of the circle or a sphere to its circumference or boundary. It is an important part of circles and spheres which is generally abbreviated as 'r'. The plural of radius is "radii" which is used when we talk about more than one radius at a time. The largest line segment in a circle or sphere joining any points lying on the opposite side of the center is the diameter, and the length of the radius is half of the length of the diameter. It can be expressed as d/2, where 'd' is the diameter of the circle or sphere. Look at the image of a circle given below showing the relationship between radius and diameter.

given:

As, CB ⊥ AC by the radius-tangent theorem.

In ΔABC, applying the Pythagoras theorem

r² + 8² = (r+ 5)²

r² + 64= r² + 25 + 10 r

64 - 25 = 10 r

39 = 10r

r= 39/10

r= 3.9 units

Hence, the radius is 3.9 units.

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