Question

D, E, F are respectively the mid-points of PQ, QR, PR of ΔPQR. The correspondence DEF⇔ ..... is similarity.Fill in the blank so that the given statement is true.

Answer 1

Given: D is the midpoint of PQ
 ⇒ PD = DQ = 1/2 PQ
But PD = DQ = FE [∵, F and E are also midpoints of PR and QR respectively]
So, EF = 1/2 PQ
or, EF/PQ = 1/2 .........(1)

E is the midpoint of QR 
⇒ QE = ER = 1/2 RQ
But QE = ER = DF [∵, D and F are also midpoints of PQ and PR respectively]
So, DF = 1/2 RQ
or, DF/RQ = 1/2 ............(2)

F is the midpoint of PR 
⇒ PF = FR = 1/2 RP
But PF = FR = DE [∵, D and E are midpoints of PQ and QR respectively]
So, DE = 1/2 RP
or, DE/RP = 1/2 ...........(3)

from equations (1) , (2) and (3),
EF/PQ = DF/RQ = DE/RP = 1/2
We can also arrange it as,
EF/PQ = DF/RQ = DE/RP
∴ The correspondence DEF ↔ RPQ is a similarity by SSS theorem.
Thus, answer is RPQ.