Tangent length: radius 5 cm, distance 13 cm

A circle has a radius of 5 cm. A point outside the circle is 13 cm from the centre. Find the length of the tangent from that point to the circle.

Step-by-step

1. 13² − 5² = 169 − 25 = 144
   The radius is perpendicular to the tangent, forming a right triangle
2. √144 = 12
   Tangent length = √(centre distance² − radius²)

Why it works

A tangent is perpendicular to the radius at the point of contact, so the radius, the tangent and the line from the centre to the outside point form a right triangle, and the tangent length follows from Pythagoras.

Common mistakes

• Subtracting the radius from the distance: 13 − 5 = 8.
• Adding the two squares before taking the square root instead of subtracting them.

Related

tangent-line parabola-vertex parabola-vertex-x2-8x-10