Find the length of the tangent segment from an external point to a circle

Key takeaway: A tangent to a circle is perpendicular to the radius at the point of tangency. That right angle is the key detail here, because it lets you use the Pythagorean Theorem cleanly.

Since $CA$ is tangent to circle $O$ at $A$, the radius $OA$ is perpendicular to $CA$.

So triangle $OAC$ is a right triangle with:

• $AC = 26$ cm
• $OA = 10$ cm
• $OC$ as the hypotenuse

Apply the Pythagorean Theorem:

$OC^2 = OA^2 + AC^2$

Substitute the values:

$OC^2 = 10^2 + 26^2$ $OC^2 = 100 + 676$ $OC^2 = 776$

Take the square root:

$OC = \sqrt{776} = 2\sqrt{194}$

So the length of $OC$ is

$\boxed{2\sqrt{194}\text{ cm}}$

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Pitfalls the pros know 👇 The usual mistake is to treat $OC$ like a radius. It is not a radius here; it is the segment from the center to the external point $C$, so it must be found with the right triangle. Another easy slip is swapping the legs and hypotenuse in the Pythagorean Theorem.

What if the problem changes? If the tangent length were different, say $AC=24$ cm while $OA=10$ cm, then

$OC^2=10^2+24^2=100+576=676,$

so $OC=26$ cm. The method stays the same; only the numbers change.

Tags: tangent, radius, Pythagorean Theorem