Chord and Tangent Intersect Externally

Here we will prove that if a chord and a tangent intersect externally then the product of the lengths of the segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.

Given: XY is a chord of a circle and TZ is a tangent that touches the circle at T. XY is produced to cut the tangent TZ at Z.

Chord and Tangent Intersect Externally

To prove: XZ × YZ = TZ2.

Construction: Join T to X and Y.




1. In ∆XTZ and ∆YTZ,

(i) ∠XZT = ∠YZT

(ii) ∠TXZ = ∠YTZ


(i) Common angle.

(ii) Angle between chord and tangent is equal to the angle in the alternate segment.

2. In ∆XTZ ∼ ∆YTZ

2. By AA criterion for similarity.

3. \(\frac{XZ}{TZ}\) = \(\frac{TZ}{YZ}\) ⟹ XZ × YZ = TZ^2 (proved).

3. Corresponding sides of similar triangles are proportional.

10th Grade Math

From Chord and Tangent Intersect Externally to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

Share this page: What’s this?

Recent Articles

  1. Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr

    Apr 20, 24 05:39 PM

    Months of the Year
    There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t…

    Read More

  2. What are Parallel Lines in Geometry? | Two Parallel Lines | Examples

    Apr 20, 24 05:29 PM

    Examples of Parallel Lines
    In parallel lines when two lines do not intersect each other at any point even if they are extended to infinity. What are parallel lines in geometry? Two lines which do not intersect each other

    Read More

  3. Perpendicular Lines | What are Perpendicular Lines in Geometry?|Symbol

    Apr 19, 24 04:01 PM

    Perpendicular Lines
    In perpendicular lines when two intersecting lines a and b are said to be perpendicular to each other if one of the angles formed by them is a right angle. In other words, Set Square Set Square If two…

    Read More

  4. Fundamental Geometrical Concepts | Point | Line | Properties of Lines

    Apr 19, 24 01:50 PM

    Point P
    The fundamental geometrical concepts depend on three basic concepts — point, line and plane. The terms cannot be precisely defined. However, the meanings of these terms are explained through examples.

    Read More

  5. What is a Polygon? | Simple Closed Curve | Triangle | Quadrilateral

    Apr 19, 24 01:22 PM

    Square - Polygon
    What is a polygon? A simple closed curve made of three or more line-segments is called a polygon. A polygon has at least three line-segments.

    Read More