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Home » Circles » Tangent Lines » Prove that the Tangent is Perpendicular to the Radius

Prove that the Tangent is Perpendicular to the Radius

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

When we introduced the topic of tangent lines we said that the tangent is perpendicular to the radius, but did not prove this. This is an important property that is used in almost every geometry problem that involves circles and a tangent line. Here we will prove that the tangent is perpendicular to the radius, at the point where it intersects the circle.

Problem

Tangent line

In circle O, l is a line tangent to the circle at point R. Prove that l⊥RO.

Strategy

Looking at the diagram, it just seems so obvious that it is true. In such cases, it is often useful to try and prove it is true using a proof by contradiction. We start by assuming that what seems to be true is false, and try to arrive at a contradiction.

A contradiction is when something impossible happens, which means our assumption is false. We assumed the opposite of what we need to prove, So if the assumption is false, it must be that what we needed to prove is true.

Let's see how this works out: We'll assume that l is not perpendicular to RO. Then there is another point on l, call it P, where OP is perpendicular to l. So triangle ΔOPR is a right triangle, with angle m∠OPR = 90°.

a right triangle with radius as hypotenuse

This means OR is the hypotenuse of that triangle, so it is the longest side of the triangle, OR>OP. Now, recall that a circle is a collection of all the points that are an equal distance - the radius- from the center of the circle.

Any point inside the circle is closer to the center than this distance, and any point outside the circle is farther than this. R is a point on the circle, and P is a point outside the circle, so OP>OR, and we have our contradiction. Our assumption that OR is not perpendicular to l is incorrect, and so we prove that the tangent is perpendicular to the radius, l⊥RO.

« HL Theorem (Hypotenuse-Leg Postulate)
Area of a Heart Shape »

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]

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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
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    • Parallelograms
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  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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