# CAT Practice : Geometry-Triangles

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Hexagon inscribed inside circle

## Chords of a circle

Q.14: A circle of radius 5 cm has chord RS at a distance of 3 units from it. Chord PQ intersects with chord RS at T such that TS = 1/3 of RT. Find minimum value of PQ.
1. 6${\surd }$3
2. 4${\surd }$3
3. 8${\surd }$3
4. 2${\surd }$3

Choice (B). 4${\surd }$3

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## Detailed Solution

OM = 3, OS = 5

MS = 4 = RM {Using Pythagoras theorem}

=> RS = 8 cms

TS = 1/3 of RT

TS = 1/4 of RS

If RS = 8 cms

TS = 2 cms

RT × TS = PT × TQ

{Intersecting Chords theorem: When there are two intersecting chords, the product of the rectangle formed by the segments of one chord is equal to the product of the rectangle formed by the segments of the other.}

6 × 2 = PT × TQ

PT × TQ = 12

By AM – GM inequality, (PT+TQ)/2 ≥ √(PT x TQ)

(PT+TQ)/2 ≥ √12

PT + TQ ≥ 2√12

=> PQ ≥ 2√12

Or PQ ≥ 4√3

Minimum PQ = 4√3

Correct Answer: 4${\surd }$3

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Geometry is probably the most vital topic as far as CAT preparation is concerned. Geometry sets the stage for Trigonometry, Cogeo and Mensuration as well.