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

New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

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.