Question

Draw a triangle. Join the mid-point of the sides of the triangle. You get 4 triangles Again join the mid-points of these triangles. Repeat this process. All the triangles drawn are similar triangles. Why?

Answer 1

Let, there is a ΔABC with the mid points D,E and F of sides AB, AC and BC.

In ΔADE and ΔABC,

D is the mid point of AB and E is the mid point of AC.
By Midpoint theorem,
DE/BC = 1/2
⇒DE/2 × BF = 1/2
 ⇒ DE/BF = 1
DE = BF ..........(1)
Similarly in ΔBFD & ΔBCA,

DF = EC = AE ............(2)
Similarly in ΔCFE & ΔCBA
EF = AD = DB .........(3)

In ΔADE & ΔBDF,
AD = DB , D is mid point
BF = DE [From equation (1)]
DF = EA [From equation (2)]
Thus, ΔADE & ΔBDF are similar to each other by SSS Similarity Rule.
⇒ ΔADE~ΔBDF
Similarly,
ΔADE~ΔEFC
ΔDBF~ΔEFC
In ΔADE & ΔDEF,
AD = EF [From equation (3)]
DE = DE
EA = DF [From equation (2)]
⇒ ΔADE~ΔDEF
Thus, all the smaller triangles are similar to each other.

here you can understand , all triangle are similar to each other and what is the reason behind it.

Answer 2

Answer:

Step-by-step explanation:Let the lengths and breadth of rectangle be l & b respectively.

Now l = b+5, perimeter = 2(l+b) = 58 cm

2(b +b +5) = 58

2b + 5 = 29

2b = 24

b = 12 and length = b +5 = 17