UPSC CDS 1 2022 Math Paper

Concept:
S.A.S. Congruence:- If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.
Median:- A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side.
Calculation:
Statement 1:
In △ADB and △EDC
AD=DE (by construction)
BD=DC (given) and
∠ADB=∠EDC (vertically opposite angles)
⇒△ADB=△EDC (by SAS congruency property)
⇒AB=CE (by the congruency of a triangle)
Now in △ACE
AC+CE>AE
⇒AC+AB>AE(∵AB=CE)
⇒AC+AB>2AD
Hence, statement 1 is incorrect.
Statement 2: In the △ABC,D,E, and F are the midpoints of sides BC,CA, and AB respectively. We know that the sum of two sides of a triangle is greater than twice the median bisecting the third side.
Hence, In △ABD, AD is a median,
⇒ AB + AC > 2AD -----(1)
Similarly, we get
⇒ BC + AC > 2CF -----(2)
⇒ BC + AB > 2BE -----(3)
On adding the above equations, we get
⇒ (AB + AC) + (BC + AC) + (BC + AB) > 2AD + 2CF + 2BE
⇒ 2(AB + BC + AC) > 2(AD + BE + CF)
⇒ AB + BC + AC > AD + BE + CF
∴ Only statement 2 is correct.