Q.3 (A)(2) Complete the following activity to prove that the sum of squares of diagonals of a rhombus is equal to the sum of the squares of the sides.[Figure: rhombus PQRS with diagonals PR and QS intersecting at T.]Given : PQRS is a rhombus. Diagona…
Q.3 (A)(2) Complete the following activity to prove that the sum of squares of diagonals of a rhombus is equal to the sum of the squares of the sides.
[Figure: rhombus PQRS with diagonals PR and QS intersecting at T.]
Given : PQRS is a rhombus. Diagonals PR and SQ intersect each other at point T.
To prove : PS2 + SR2 + QR2 + PQ2 = PR2 + QS2
Activity :
Diagonals of a rhombus bisect each other.
In Δ PQS, PT is the median and in Δ QRS, RT is the median.
∴ by Apollonius theorem,
PQ2 + PS2 = ____ + 2QT2 ................. (I)
QR2 + SR2 = ____ + 2QT2 ................. (II)
adding (I) and (II),
PQ2 + PS2 + QR2 + SR2 = 2 (PT2 + ____) + 4QT2
= 2 (PT2 + ____) + 4QT2 .............. (RT = PT)
= 4PT2 + 4QT2
= (____)2 + (2QT)2
∴ PQ2 + PS2 + QR2 + SR2 = PR2 + ____ .