Geometry Problem: Finding the Circle Radius

This problem involves finding the radius of a circle using the relationship between the distance from the center to an external point and the length of the tangent from that point.

Applying the Tangent-Radius Theorem

According to the tangent-radius theorem, the radius drawn to the point of tangency is perpendicular to the tangent line at that point. This forms a right-angled triangle where:

• The distance from the center of the circle to the external point (P) is the hypotenuse.
• The length of the tangent from point P to the circle is one leg of the triangle.
• The radius of the circle is the other leg of the triangle.

Using the Pythagorean Theorem

Let:

• Distance from the center to point P = $d = 13$ cm
• Length of the tangent from P = $l = 12$ cm
• Radius of the circle = $r$

Applying the Pythagorean theorem ($hypotenuse^2 = leg1^2 + leg2^2$):

$d^2 = l^2 + r^2$

Substitute the given values:

$13^2 = 12^2 + r^2$

$169 = 144 + r^2$

Solve for $r^2$:

$r^2 = 169 - 144$

$r^2 = 25$

Solve for $r$:

$r = \sqrt{25}$

$r = 5 \text{ cm}$

Conclusion

The radius of the circle is 5 cm.