Given : ABCD is a rhombus, DPR and CBR are straight lines. Prove that : DP $\times$ CR = DC $\times$ PR.

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Solution

Given: ABCD is a rhombus.A line DPR meets AC at P and CB produced at R.
To Prove:DP×CR=DC×PR
Proof:
In △APD and△CPR,
∠APD=∠CPR [vertically opposite angles]
∠DAP=∠PCR [ alternate interior angles as , AD∥BC and AC is a transversal]
Thus, △APD~△CPR [ AA criterion of similarity]
Since,corresponding sides of similar triangles are proportional
$fraction numerator D P over denominator P R end fraction equals fraction numerator A D over denominator C R end fraction fraction numerator D P over denominator P R end fraction equals fraction numerator D C over denominator C R end fraction space space space open parentheses A D equals D C space comma a s space A B C D space i s space r h o m b u s close parentheses h e n c e space D P cross times C R equals D C cross times P R$

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