Question

PQRS is a square of side 8 cm. T and U and respectively the mid points of PS and QR. If TU and QS intersect at 0, then ar(ΔOTS) = ________.

Answer 1

Given:
PQRS is a square of side 8 cm.
T and U are respectively the mid-points of PS and QR
TU and QS intersect at O



In ΔQOU and ΔOTS,
∠QOU = ∠TOS (vertically opposite angles)
∠OQU = ∠OST (alternate angles)
QU = TS (mid-points of sides of a square)

By AAS property,
ΔQOU ≅ ΔOTS

Thus, OU = OT (by CPCT)
⇒ OU + OT = PQ = 8 cm
⇒ OU = OT = 4 cm ...(1)

Also, TS = 4 cm (T is the mid-point of PS) ...(2)

ar(ΔOTS) = $\frac{1}{2}$× base × height
= $\frac{1}{2}$× TS × OT
= $\frac{1}{2}$× 4 × 4 (From (1) and (2))
= 8 cm²
​

Hence, ar(ΔOTS) = 8 cm².