Prove that sum of any two sides of a triangle is greater than twice the median with respect to the third side.

Which of the following statements are true (T) and which are false (F)? (i) Sum of the three sides of a triangle is less than the sum of its three altitudes. (ii) Sum of any two sides of a triangle is greater than twice the median drawn to the third side. (iii) Sum of any two sides of a triangle is greater than the third side. (iv) Difference of any two sides of a triangle is equal to the third side. (v) If two angles of a triangle are unequal, then the greater angle has the larger side opposite to it. (vi) Of all the line segments that can be drawn from a point to a line not containing it, the perpendicular line segment is the shortest one.

Prove that any two sides of a triangle are together greater than twice the median drawn to the third side. GIVEN : triangle A B C in which A D is a median. PROVE : A B+A C >2A D CONSTRUCTION : Produce A D to E such that A D=D E . Join E C .

Prove that any two sides of a triangle are together greater than twice the median drawn to the third side.

Theorem 7.8 : The sum of any two sides of a triangle is greater than the third side.

Prove that the perimeter of a triangle is greater than the sum of its three medians.

If two sides of a triangle are hati+2hatj and hati+hatk , then find the length of the third side.

Prove that the sum of the sides of a quadrilateral is greater than twice of one of its diagonal.

If two side of a triangle are hat i+2 hat j and hat i+ hat k , then find the length of the third side.

Show that the difference of any two sides of a triangle is less than the third side.

Theorem 6.2 : If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.