Shortcut Trick
When a diameter is perpendicular to a chord, it bisects the chord: RT = TS = 16 ÷ 2 = 8 cm.
According to the Intersecting Chords Theorem, RT × TS = PT × TQ.
⇒ 8 × 8 = PT × 4
⇒ PT = 64 ÷ 4 = 16 cm.
⇒ Diameter PQ = PT + TQ = 16 + 4 = 20 cm.
∴ The correct answer is 20 cm.
Alternate Method
Given: Chord RS = 16 cm, PQ ⊥ RS at T, QT = 4 cm.
Formula Used: Pythagoras Theorem: Hypotenuse2 = Base2 + Perpendicular2
⇒ Let the radius of the circle be r cm.
⇒ In ▵OTR, OR is the radius, so OR = r.
⇒ Since PQ is the diameter, OQ = r. Given QT = 4 cm, OT = OQ − QT = r − 4.
⇒ RS is bisected by the diameter at T, so RT = 16 ÷ 2 = 8 cm.
⇒ By Pythagoras Theorem in ▵OTR: OR2 = OT2 + RT2
⇒ r2 = (r − 4)2 + 82
⇒ r2 = r2 − 8r + 16 + 64
⇒ 8r = 80
⇒ r = 10 cm
⇒ Diameter = 2 × r = 2 × 10 = 20 cm.
∴ The correct answer is 20 cm.
Additional Information
Perpendicular from Center
A perpendicular drawn from the center of a circle to a chord always bisects the chord into two equal segments.
Intersecting Chords Theorem
For two chords intersecting at a point T inside a circle, the product of the segments of one chord equals the product of the segments of the other: AT × TB = CT × TD.
Radius and Diameter
The diameter is the longest chord of a circle and its length is exactly twice the length of the radius (D = 2r).