Alternate segment theorem

Summary

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A line that touches a circle at only one point is known as the tangent to the circle. The point that is common to the tangent and the circle is known as the point of contact. The radius or the diameter of the circle drawn at the point of contact is always perpendicular to the tangent.

A chord at the point of contact divides the circle into two segments: The major segment and the minor segment.

Alternate segment theorem: “If a chord is drawn through the point of contact of a tangent to a circle, then the angles that this chord makes with the given tangent are respectively equal to the angles formed in the corresponding alternate segments.” The converse of this theorem also holds good.

This can be proved mathematically.

Questions & Answers

1 . Perimeter of a right triangle is equal to the sum of the diameter
ABC is a right angled triangle right angled at B. Hence AC is diameter of the circumcircle as angle in a semi circle is a right angle. O...
2 . Prove that any four vertices of a regular pentagon are concyclic
Given ABCDE is a regular pentagon That is AB = BC = CD = DE = AE Recall that the sum of angles in a regular pentagon is 540° Hence each...
3 . ABC is right angle with angel B=90".
Since BC is diameter of the circle, ∠BDC = 90° [Since angle in a semi circle is a right angle] Consider Δ’s ABC and ADB ∠B = ∠ADB = 90° ...
4 . form an external point P tangent PA and PB are drawn to a circle .If CD is the tangent to the circle at a point E and PA=15cm,find the perimeter of the triangle PCD.
Sol : Perimeter of ΔPCD = 2 x AP.                             =2 x 15 = 30 cm.
5 . Prove that ΔPRS ~ΔPTQ. If PQ = 4cm ,PT=3cm , ST= 5cm.
Sol: PQ = 4 cm, PT = 3 cm , ST = 5 cm (i) PQ x PR = PT x PS 4 x PR = 3 x 8 PR = 6 cm QR = PR - PQ QR = 6 - 4 QR = 2 cm (ii) Area ...