• Skip to primary navigation
  • Skip to main content
  • Skip to primary sidebar
Geometry Help
  • About
  • Privacy Policy
  • Contact Me
  • Terms of Service
  • Accessibility Statement
menu icon
go to homepage
search icon
Homepage link
  • About
  • Privacy Policy
  • Contact Me
  • Terms of Service
  • Accessibility Statement
×
Home » Circles » Tangent Lines » Tangent Line to a Circle

Tangent Line to a Circle

Last updated: Sep 23, 2022 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

A line tangent to a circle is a line from a point outside the circle that touches the circle at exactly one point. At that point, the tangent line is perpendicular to the circle's radius and diameter, and this property is often used in high school geometry problems involving tangent lines.

Problem

Line AB is tangent to circle O at point B, and the line segment AB measures 12 units. The line AO, from point A to the circle's center, intersects the circle at point C, and the line segment AC measures 8 units. Find the circle's radius.

tangent line to a circle

Strategy

The problem features a tangent line, which is a hint to use the property of tangent lines to a circle: that line is perpendicular to the circle's radius at the point where it touches the circle.

So ∠OBA=90° and ΔOBA is a right triangle, whose long leg's length we know (AB=12). The circle's radius, which is what we need to find, is the short leg of the right triangle (OB).

Next, we note that the hypotenuse, AO, is made up of a segment whose length we know (AC=8) and another radius - OC. We can call the radius x, and solve for x using the Pythagorean Theorem.

Solution

(1) AB⊥OB //given, AB is tangent to circle O, a tangent line is perpendicular to the radius
(2) ∠OBA=90° //Definition of perpendicular line
(3) ΔOBA is a right triangle //Definition of right triangle
(4) AO2=AB2+OB2 //Pythagorean Theorem
(5) AB=12 //Given
(6) r=OB=x
(7)AO=x+8
(8) (x+8)2=122+x2 //substitute for AB, OB and AO
(9) x2+16x+64=144+x2 //expand (x+8)2
(10) 16x+64=144 //Subtract x2 from both sides
(11) 16x=80 //Subtract 64 from both sides
(12) x=5=r //Divide both sides by 16, x is hte radius

« Congruent Triangles in a Circle
Two Lines Parallel to a Third are Parallel to Each Other »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

Primary Sidebar

About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
    • Arcs, Angles, and Sectors
    • Chords
    • Inscribed Shapes
    • Tangent Lines
  • Lines and Angles
    • Intersecting Lines and Angles
    • Parallel Lines
    • Perpendicular lines
  • Pentagons and Hexagons
  • Perimeter of Geometric Shapes
  • Polygons
  • Quadrangles
    • Kites (Deltoids)
    • Parallelograms
    • Rectangles
    • Rhombus
    • Squares
    • Trapezoids
  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

By accessing or using this website, you agree to abide by the Terms of Service and Privacy Policy.


Copyright © 2023