Question

Example 7: In A ABC, D, E and F are respectively
the mid-points of sides AB, BC and CA
(see Fig. 8.27). Show that A ABC is divided into four
congruent triangles by joining D, E and F.​

Answer 1

$\huge\sf \underline \red{Answer \: }$

$\sf \underline{ \therefore \triangle \: DBF = \triangle DFE = \triangle \: CEF = \triangle ADF}$

• This all four congruent triangles

$\huge\sf \underline \blue{To \: proof : }$

• ABC is divided into four congruent triangles

$\huge \sf \underline \pink{Solution : }$

$\sf \underline{Given : }$

• ABC is a triangle D , E , F are respectively the mid points of of sides AB, BC and CA.

D and F are mid points of sides AB and AC of ABC so,

$\: \: \: \: \: \: \sf \underline{DF \: ll \: BC}$

• line segment is joining the mid points of 2 sides of a triangle is parallel to the third side

$\sf \underline{so}$

$\sf{DE \: ll \: AC \: EF \: ll \: AB}$

$\sf \underline{Now \: DBEF \: }$

$\sf{DF \: ll \: BE(DF \: ll \: BC) \: parallel \: lines \: are \: parallel}$

$\sf{DB \: ll \: EF (AB \: ll \: EF)parallel \: lines \: are \: parallel}$

• Both pairs are opposite sides are parallel

$\sf \underline{ DBEF \:is \: a \: parallelogram }$

$\sf {Now \: DBEF \: is \: a \: paralleogram \: DE \: is \: a \: diagonal}$

$\sf{ \triangle \: DBE = \triangle \: DFE}$

• Diagonal of a parallelogram will be divide into two congruent triangle

• This is the first equation (1)

so,

$\sf{DFCE \: is \: a \: parallelogram \: \triangle \: DFE = \triangle \: CEF}$

• This is second equation (2)

$\sf{ADEF \: is \: a \: parallelogram \: \triangle \: ADF = \triangle ADE}$

• This is the third equation (3)

$\sf \underline{Now \: from \: \: equation (1) (2) (3) we \: get }$

$\sf{ \triangle \: DBF = \triangle DFE = \triangle \: CEF = \triangle ADF}$

• These are the four congruent triangles

$\sf \underline{ \therefore \triangle \: DBF = \triangle DFE = \triangle \: CEF = \triangle ADF}$

Answer 2

Answer:

This all four congruent triangles

ABC is divided into four congruent triangles

ABC is a triangle D , E , F are respectively the mid points of of sides AB, BC and CA.

D and F are mid points of sides AB and AC of ABC so,

line segment is joining the mid points of 2 sides of a triangle is parallel to the third side

Both pairs are opposite sides are parallel

Diagonal of a parallelogram will be divide into two congruent triangle

This is the first equation (1)

so,

This is second equation (2)

This is the third equation (3)

These are the four congruent triangles

Step-by-step explanation:

Hope this answer will help you✌