Question

ABCD is a rhombus, DPR and CBR are straight lines.

Prove that : DP × CR = DC × PR.​

Answer 1

ABCD is a rhombus. DPR and CBR are straight lines.

∴  AD∥CR

In △APD and △CPR

⇒  ∠APD=∠CPR               [ Vertically opposite angles ]

⇒  ∠DAP=∠PCR               [ Alternate angles ]

∴  △APD∼△CPR               [ By AA criterion ]

Since, corresponding sides of similar triangles are proportional.

PR /DP  =  CR /AD  =  CP /AP

​⇒    PR /DP  =  CR /AD

​⇒    PR /DP  =  CR /DC   [ Since, DC=AD, as ABCD is a rhombus ]

⇒  DP×CR=DC×PR

Hence, proved

Answer 2

Step-by-step explanation:

ABCD is a rhombus. DPR and CBR are straight lines.

∴ AD∥CR

In △APD and △CPR

⇒ ∠APD=∠CPR [ Vertically opposite angles ]

⇒ ∠DAP=∠PCR [ Alternate angles ]

∴ △APD∼△CPR [ By AA criterion ]

Since, corresponding sides of similar triangles are proportional.

PR/DP= CR/AD= CP/AP

⇒PR/DP= CR/AD

⇒PR/DP=CR/DC [ Since, DC=AD, as ABCD is a rhombus ]

⇒ DP×CR=DC×PR [PROVED]