Let D,E and F be the midpoints of the sides BC,CA and AB respectively of a triangle ABC. Triangle DEF is congruent to which of the following triangles? 1. AEF 2. FBD 3. EDC Select the correct answer using the code given below.
Concept: Mid Point Theorem:- The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side. S.S.S. Congruence:- If all the three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles are said to be congruent. Calculation:
According to the midpoint theorem, ⇒DE=‌
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2
AB‌ and ‌DE|AB ⇒DE=AF=BF . . . (i) Similarly, ⇒FD=‌
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AC and FD|AC ⇒FD=AE=EC . . . (2) ⇒EF=‌
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2
BC and EF|BC ⇒EF=BD=DC . . . (3) ⇒ In △AFE and △DFE ⇒AF=DE‌‌[from(1)] ⇒AE=DF‌‌[ from (2)] ⇒EF=EF‌‌ [common] ⇒△AFE≅△DEF‌‌ [by SSS congruency] ⇒ In △FBD and △DEF ⇒DF=DF‌‌ [common] ⇒BF=DE‌‌[from(1)] ⇒BD=EF‌‌[ from (3)] ⇒△FBD≅△DEF[ by SSS congruency] ⇒ In △EDC and △DEF ⇒DE=DE‌‌[‌ common ‌] ⇒CE=DF‌‌[‌ from ‌(2)] ⇒CD=EF‌‌[from(3)] ⇒△EDC≅△DEF[‌ by SSS congruency ‌] ∴△DEF is congruent to △AFE,△FBD, and △EDC.