Question

Find the value of ‘x’ if root prove that (64/125) -2/3 find two rational numbers between 1/2 and 2/3 find two irrational numbers between 2 and 3 factorise: x 2 -2x + 1 prove that every line segment has one and only one mid-point show taht the angles of an equilateral triangle are 60 degree each if a + b + c =7 and ab +bc + ca = 6 then find the value of a 2 +b 2 + c 2 prove that sum of three angles of a triangle is 180 degree represent root 3 on the real number line if 7-3 root2= a+ b root 2 7 + 3root2

Answer 1

let there are two mid points of a line segment ac i.e.  b and d
hence ab=bc and ad =dc
ac=ab+bc
and ac =ad+dc 
so ad+dc=ab + bc
then ad+ad=ab+ab as ab=bc and ad=dc
2ab =2ad
ab=ad

Answer 2

Let us assume that the line segment AB has 2 midpoints C and D. Then, AC + BC = AB (C is the midpoint) ----> 1 And, AD +BD = AB (D is the midpoint ) ----> 2 From 1 and 2 we get, AC + BC = AD + BD AC - BD = AD - BC -CD = CD 2CD = 0 CD = 0 This is possible only if the points C and D coincide. Therefore a line segment has one and only one midpoint. Hence Proved.