2016 · Q26
Let A B C and A' B' C' be two triangles in which A B > A' B', B C > B' C' and C A > C' A'. Let D, E and F be the midpoints of the sides B C, C A and A B respectively. Let D', E' and F' be the midpoints of the sides B' C', C' A' and A' B' respectively. Consider the following statements :
Statement IAD > A'D', BE > B'E' and CF > C'F' are always true.
Statement II(AB²+BC²+CA²)/(AD²+BE²+CF²) = (A'B'²+B'C'²+C'A'²)/(A'D'²+B'E'²+C'F'²)Which one of the following is correct in respect of the above statements?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- ABoth Statement I and Statement II are true and Statement II is the correct explanation of Statement I
- BBoth Statement I and Statement II are true but Statement II is not the correct explanation of Statement I
- CStatement I is true but Statement II is false
- DStatement I is false but Statement II is true
Explanation
By Apollonius's theorem the sum of the squared medians is 3/4 of the sum of the squared sides for every triangle, so both ratios in Statement II equal 4/3 — Statement II is true. Statement I is false: larger sides do not always give larger medians (counterexamples exist). Hence Statement I is false but Statement II is true, option D.