Option 1 : 3/4

**Given:**

ΔPQR is an equilateral triangle which is inscribed in a circle and PQ = 10 cm.

Bisector of ∠PQR meets PR at a point of T and meets the circle at a point of S.

**Concept used:**

Height of a equilateral triangle = √3/2 × side

Properties of chords:- If two chords of a circle intersect inside the circle, then the product of the measures of the segments of one chord is equal to the product of the measures of the segments of the other chord.

**Calculation:**

Here in the fig, ΔPQR is an equilateral triangle.

So, PQ = QR = PR = 10 cm

QT = Height = √3/2 × 10

⇒ QT = 5√3

Now, from the properties of chord,

PT × TR = QT × TS

⇒ TS = 25/5√3

⇒ TS = 5/√3

⇒ TS = 5√3/3

Now,

QS = QT + TS

⇒ QS = 5√3/3 + 5√3

⇒ QS = 20√3/3

So, QT/QS = 15/20 = 3/4

**∴ required value is 3/4**