Prove that the perimeter of a triangle is greater than the sum of the three medians. GIVEN : A ` A B C` in which `A D ,B E` and `C F` are its medians. TO PROVE : `A B+B C+A C > A D+B E+C F`

Show that the sum of the three altitudes of a triangle is less than the sum of three sides of the triangle. GIVEN :triangle A B C in which A D_|_B C ,B E_|_A C and C F_|_A Bdot PROVE : A D+B E+C F < A B+B C+A C

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 .

Parallelogram A B C D and rectangle A B E F have the same base A B and also have equal areas. Show that the perimeter of the parallelogram is greater than that of the rectangle. GIVEN : A ||^(gm)A B C D and a rectangle A B E F with the same base A B and equal areas. TO PROVE : Perimeter of ||^(gm)A B C D > Perimeter of rectangle ABEF i.e. A B+B C+C D+A D > A B+B E+E F+A F

A B C is a triangle in which D is the mid-point of BC and E is the mid-point of A D . Prove that area of triangle B E D=1/4area \ of triangle A B C . GIVEN : A triangle A B C ,D is the mid-point of B C and E is the mid-point of the median A D . TO PROVE : a r( triangle B E D)=1/4a r(triangle A B C)dot

The sum of any two sides of a triangle is greater than the third side. GIVE : A triangle A B C TO PROVE : A B+A C > B C ,A B+B C > A C and B C+A C > A B CONSTRUCTION : Produce side B A to D such that A D=A Cdot Join C D .

In Figure, A D , and B E are medians of A B C and B E || D Fdot Prove that C F=1/4A Cdot

A B C is an isosceles triangle in which A B=A C,B E\ a n d\ C F are its two medians. Show that B E=C Fdot

If any two angles and a non-included side of one triangle are equal to the corresponding angles and side of another triangle, then the two triangles are congruent. GIVEN : Two s A B C and D E F such that /_A=/_D ,/_B=/_E ,B C=E F TO PROVE : A B C~=D E F Figure

Two triangles are congruent if two sides and the included angle of one are equal to the corresponding sides and the included angle of the other triangle. GIVEN : Two triangles A B C and D E F such that A B=D E ,A C=D F and /_A=/_D PROVE : A B C~= D E F Figure

In a A B C ,\ \ B D and C E are the altitudes. Prove that A D B and A E C are similar. Is C D B ~ B E C ?