In the given figure, ABC and CEF are two triangles where BA is parallel to CE and AF : AC = 5 : 8. [3 MARKS] i) Prove that ΔADF∼ΔCEF ii) Find AD, if CE = 6 cm iii) If DF is parallel to BC, find area of ΔADF : area of ΔABC
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Solution
Each part: 1 Mark
i) In ΔADF and ΔCFE, ∠DAF=∠ECF [Alternate angles] ∠AFD=∠CFE [Vertically opposite angles] ∴∠ADF=∠CEF ΔADF∼ΔCEF [By AAA similarity] Hence, proved
ii) Since the corresponding sides of similar triangles are proportional. ∴ In ΔADF and ΔCEF, ADCE=AFFC⇒AD6=58−5 ⇒AD6=53 ⇒AD=303=10cm
iii) If DF is parallel to BC, then ΔADF∼ΔABC Now, ar(ΔADF)ar(ΔABC)=AF2AC2 =5282=2564