`D ,\ E ,\ F` are the mid-points of the sides `B C ,\ C A` and `A B` respectively of a ` A B C` . Determine the ratio of the areas of ` D E F` and ` A B C` .

If D ,E ,F are the mid-points of the sides B C ,C Aa n dA B respectively of a triangle A B C , prove by vector method that A r e aof D E F=1/4(a r e aof A B C)dot

D ,\ E ,\ F are the mid-point of the sides B C ,\ C A\ a n d\ A B respectively of A B Cdot Then D E F is congruent to triangle. A B C (b) AEF (c) B F D ,\ C D E (d) A F E ,\ B F D ,\ C D E

In a A B C ,\ D ,\ E ,\ F are the mi-points of sides B C ,\ C A\ a n d\ A B respectively. If a r( A B C)=16 c m^2, then a r\ (t r a p e z i u m\ F B C E)= 4\ c m^2 (b) 8\ c m^2 (c) 12 c m^2 (d) 10 c m^2

Let A B C be an isosceles triangle in which A B=A Cdot If D ,\ E ,\ F be the mid-points of the sides B C ,\ C A\ a n d\ A B respectively, show that the segemtn A D\ a n d\ E F bisect each other at right angles.

If O is a point in space, A B C is a triangle and D , E , F are the mid-points of the sides B C ,C A and A B respectively of the triangle, prove that vec O A + vec O B+ vec O C= vec O D+ vec O E+ vec O Fdot

D ,\ E and F are the points on sides B C ,\ C A and A B respectively of A B C such that A D bisects /_A ,\ \ B E bisects /_B and C F bisects /_C . If A B=5c m ,\ \ B C=8c m and C A=4c m , determine A F ,\ C E and B D .

In a A B C ,\ E\ a n d\ F are the mid-points of A C\ a n d\ A B respectively. The altitude A P\ to\ B C intersects F E\ at Qdot Prove that A Q=Q P

A B C D is a trapezium with parallel sides A B=a\ a n d\ D C=bdot If E\ a n d\ F are mid-points of non-parallel sides A D\ a n d\ B C respectively, then the ratio of areas of quadrilaterals A B F E\ a n d\ E F C D is: a : b (b) (a+3b):(3a+b) (3a+b):(a+3b) (d) (3a+b):(3a+b)

D, E and F are respectively the mid-points of the sides BC, CA and AB of aDeltaA B C . Show that (i) BDEF is a parallelogram. (ii) a r" "(D E F)=1/4a r" "(A B C) (iii) a r" "(B D E F)=1/2a r" "(A B C)