Point B is a point of tangency. Find the radius r of OC.

Understand the Problem

The question is asking for the radius of a circle given a point of tangency and some measurements of line segments related to the tangential point. We need to use the properties of tangents and the geometry of the circle to find the radius.

Answer

The radius $ r $ is $ 10.5 $.

Steps to Solve

Identify the Tangential Point and Segments

Given point ( B ) as the point of tangency implies that the line segment ( OB ) is perpendicular to the tangent line ( AB ) at point ( B ). Thus, ( AB ) is a tangent to the circle at point ( B ).

Set Up the Right Triangle

We can observe a right triangle formed by points ( A ), ( B ), and ( C ). The sides are:

( AB = 14 ) (the tangent segment)
( BC = r ) (the radius)
( AC = r + 7 ) (the height from point ( A ) to ( C ))

Apply the Pythagorean Theorem

Using the Pythagorean theorem for triangle ( ABC ): $$ AB^2 + BC^2 = AC^2 $$ Substituting the known values: $$ 14^2 + r^2 = (r + 7)^2 $$

Expand and Simplify

Now expand the right side: $$ 14^2 + r^2 = r^2 + 14r + 49 $$ This leads to: $$ 196 + r^2 = r^2 + 14r + 49 $$

Solve the Equation

Subtract ( r^2 ) from both sides: $$ 196 = 14r + 49 $$ Subtract 49 from both sides: $$ 147 = 14r $$ Now, divide both sides by 14: $$ r = \frac{147}{14} = 10.5 $$

The radius ( r ) of ( OC ) is ( 10.5 ).

More Information

The radius of a circle is the distance from the center to any point on the circle. In this case, point ( B ) being a point of tangency signifies that the radius at ( C ) is perpendicular to the tangent at ( B ), forming a right triangle with the tangent segment and the difference in height from point ( A ) to ( C ).

Tips

Forgetting to use the Pythagorean theorem correctly for right triangles.
Failing to set up the equation with correct values for tangent length and distances.
Making algebraic mistakes when expanding expressions.

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