question_answer

In the given figure (not to scale) \[AB=CD\] and \[BC=DE\] What type of a triangle is \[\Delta ACE\]?

A)  Isosceles triangle                  

B)  Equilateral triangle

C)  Scalene right - angled triangle      

D)  Right - angled isosceles triangle

Correct Answer: B

Solution :

(b): This is again an excellent test of mind for the students: From given data, \[\Delta ABC\cong \Delta CDE\] \[\Rightarrow \,\,AC=EC\,\,\,\Rightarrow \,\,\,\,\angle p=\angle q\] \[\angle y=\angle m;\,\,\,\angle x=180{}^\circ -60{}^\circ -m=120{}^\circ -m\] At point C:  \[\angle ACE=180{}^\circ -x-y\] \[=180{}^\circ -(120-m)-m=60{}^\circ \] Also \[\angle ACE=180{}^\circ -\angle p-\angle q=180{}^\circ -2\angle p=60{}^\circ \] \[\Rightarrow \angle p=\angle q=60{}^\circ \Rightarrow \Delta \,\,\Alpha CE\] is equilateral \[\Delta \,\,le\].