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ML Aggarwal Solutions Class 9 Mathematics Solutions for Mid Point Theorem Exercise 11 in Chapter 11 - Mid Point Theorem

Question 2 Mid Point Theorem Exercise 11

Prove that the four triangles formed by joining in pairs the mid-points of the sides C of a triangle are

congruent to each other

Answer:

It is given that

In Δ ABC

D, E, and F are the mid-points of AB, BC, and CA

Now join DE, EF, and FD

To find:

Δ ADF ≅ Δ DBE ≅ Δ ECF ≅ Δ DEF

M L Aggarwal - Understanding ICSE Mathematics - Class 9 chapter Mid Point Theorem Question 2 Solution image

To prove:

In Δ ABC

D and E are the mid-points of AB and BC

DE || AC or FC

Similarly DF || EC

DECF is a parallelogram

We know that

Diagonal FE divides the parallelogram DECF in two congruent triangles DEF and CEF

Δ DEF ≅ Δ ECF …… (1)

Here we can prove that

Δ DBE ≅ Δ DEF …. (2)

Δ DEF ≅ Δ ADF ……. (3)

Using equation (1), (2) and (3)

Δ ADF ≅ Δ DBE ≅ Δ ECF ≅ Δ DEF

Video transcript
"hey guys welcome to lido q a video and we need your little tutor bringing you this question on your screen prove that the four triangles formed by joining in pairs the midpoints of the sides of a triangle are congruent to each other so let us see what the question means right we draw a triangle yeah now after we draw the triangle let us mark the midpoints and join them so this is given to us right so a b c d e n f we have to prove that triangle a d e is congruent to triangle b d f is congruent to triangle f e c is congruent to triangle d e f how do we do that now in the big triangle right d and e are midpoints of a b and ac respectively therefore by midpoint formula we can say that this portion ef right so in this triangle ef is parallel to a b df is parallel to ac right so ef is parallel to a b or e d and df is parallel to ae therefore we can say that this portion a d f e right this portion is a parallelogram a d f e is a parallelogram now we know that diagonal of a parallelogram divides [Music] it into two congruent triangles [Music] right therefore triangle ade is congruent to triangle d e f similarly we can prove triangle a d e right sorry triangle ade is parallel to this then we have triangle ade is parallel to triangle def okay then also we have triangle the other four triangles are also parallel like congruent triangle ade is congruent to triangle what are the parallel okay triangle sorry not triangle ade but triangle bdf is congruent to triangle def and triangle cef is also congruent to triangle def therefore triangle ade is congruent to triangle dbf is congruent to triangle def is congruent to triangle f e c isn't that easy guys right so if you have a doubt please leave a comment below do like the video and subscribe to our channel i'll see you in our next video until then bye guys keep practicing"
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