Home
Class 9
MATHS
In the adjoining figure D, E and F are t...

In the adjoining figure D, E and F are the mid-points of the sides BC, CA and AB respectively of `Delta ABC`. Prove that:
(i) `square BDEF` is a parallelogram
(ii) area of `Delta DEF = (1)/(4) xx " area of " Delta ABC`
(iii) area of `square BDEF = (1)/(2) xx " area of " Delta ABC`

Text Solution

Verified by Experts

(i) In `Delta ABC`
Since, F is the mid-point of AB and E is the mid-point of AC
`{:( :." "FE ||BC),(rArr" "FE ||BD),("Similarly "ED||FB):}` (mid - point theorem)
`:. Square BDEF` is a parallelogram (`:.` pairs of opposite sides are parallel)
(ii) Since, `square BDEF` is a parallelogram and FD is its diagonal
`:.` area of `Delta FBD = " area of " Delta DEF`..(1)
Similarly, we can prove that
area of `Delta CDE = " area of " Delta DEF`...(2)
and area of `Delta AFE = " area of " Delta DEF`...(3)
`:. 4 xx " area of " Delta DEF = " area of "Delta DEF + " area of " Delta DEF + " area of "Delta DEF + " area of "Delta DEF`
`= " area of " Delta DEF + " area of "Delta AEF + " area of "Delta FBD + " area of "Delta CDE`
= area of `Delta ABC`
`rArr " area of " Delta DEF = (1)/(4) xx " area of " Delta ABC`
(iii) Now, area of `square BDEF = 2 xx " area of " Delta ABC`
(`:.` diagonal divides the parallelogram into two equal areas)
`= 2 xx (1)/(4) xx " area of " Delta ABC`
`= (1)/(2) xx " area of "Delta ABC`
Promotional Banner

Topper's Solved these Questions

  • AREA OF PARALLELOGRAMS AND TRIANGLES

    NAGEEN PRAKASHAN|Exercise Problems From NCERT/exemplar|12 Videos

  • AREA OF PARALLELOGRAMS AND TRIANGLES

    NAGEEN PRAKASHAN|Exercise Exercise|34 Videos

  • CIRCLE

    NAGEEN PRAKASHAN|Exercise Revision Exercise (long Answer Questions )|5 Videos

Similar Questions

Explore conceptually related problems

In the adjoining figure D, E and F are the mid-points of the sides BC, AC and AB respectively. triangle DEF is congruent to triangle :

D, E and F are the mid-points of the sides BC, CA and AB of a triangle ABC. Show that BDEF is a parallelogram

D, E and F are the mid-points of the sides BC, CA and AB, respectively of and equilateral Delta ABC. Show that Delta DEF is also an equilateral triangle.

D,E and F are respectively the mid-points of the sides BC,CA and AB of a /_ABC. Show that (i) BDEF is a parallelogram

If D, E and F are the mid points of BC, CA and AB respectively of the triangle ABC then the ratio of aria parallelogram DEFB and area of the trapezium CAFD is ,

In the adjoning figure, D and E are the points on the sides AB and AC respectively of Delta ABC and area of Delta BCE = " area of " Delta BCD . Prove that DE ||BC

D, E, F are the midpoints of the sides BC, CA and AB respectively of triangle ABC. Prove that (i) BDEF is a ||gm, (ii) ar(triangleDEF)=(1)/(4) ar(triangleABC) and (iii) ar("||gm BDEF") = (1)/(2)ar(triangleABC)