In the given figure, in Δ PQR, PQ = 8 cm, ∠P = 90° and median PS is perpendicular to median QT at X. What is the length of QT? 

Given:

In the given figure, in Δ PQR, PQ = 8 cm, ∠P = 90° and median PS is perpendicular to median QT at X.

Formula used:

When two medians intersect at a point called the "centroid" which divides each median in a ratio of 2:1.

Calculation:

When two medians intersect at a point called the "centroid" which divides each median in a ratio of 2:1.

So, PX : XS = 2a : a

QX : XT = 2b : b

Now, using Pythagoras theorem

In Δ PQX,

PX² + QX² = PQ²

⇒ (2a)² + (2b)² = 8²

⇒ 4a² + 4b² = 64     --- (1)

In Δ PXT,

PX2 + XT2 = PT2

⇒ (2a)2 + b2  = PT2

⇒ 4a2 + b2 = PT2    ---- (2)

In Δ PQT,

QT² = PQ² + PT²

⇒ QT² = 8² + 4a² + b²

⇒ (2b + b)² = 64 + (64 - 4b²) + b²   (from equation (1))

⇒ (3b)² = 64 + 64 - 4b² + b²

⇒ 9b² = 128 - 3b²

⇒ 12b² = 128

⇒ b² = 128/12 = 32/3

⇒ b = √(32/3) = √(32 × 3)/ (3 × 3) = √(96/9) = √96 / 3

So, QT = 3b = 3 × √96 / 3 = √96 = 4√6 cm

Therefore, the correct answer is option (3).