In the given figure all triangles are equilateral and AB = 8 units. Other triangles have been formed by taking the mid points of the sides. Then the perimeter of the figure is ____.

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In the given figure all triangles are equilateral and AB = 8 units. Other triangles have been formed by taking the mid points of the sides. Then the perimeter of the figure is 45 units.
Given figure is,
Question Image

All triangles are equilateral and AB = 8 units.
Mark the given figure.
Question Image

In

Δ

ABC,
AB = BC = CA = 8 units.
We know that triangles formed by mid points of the sides is also an equilateral triangle.
Since E is the midpoint of AB.

2 EA = AB

\Rightarrow EA = \frac{AB}{2}

\Rightarrow EA = \frac{8}{2}

\Rightarrow EA = 4

∴

Δ

ADE is also an equilateral triangle.
∴

AE = BE = EA = 4

units
In

Δ

BOT,
O is the midpoint of BC.

2 BO = BC

\Rightarrow BO = \frac{BC}{2}

\Rightarrow BO = \frac{8}{2}

\Rightarrow BO = 4 units

Δ

BOT is also an equilateral triangle.
∴

BO = OT = BT = 4

units
Similarly,

Δ

QPC is also an equilateral triangle.
∴

QC = PC = PQ = 4

units.
Now,

Δ

DIF is also an equilateral triangle and F is the midpoint of DE.
∴

2 DF = DE

\Rightarrow DF = \frac{DE}{2}

\Rightarrow DF = \frac{4}{2}

\Rightarrow DF = 2 units

∴

DF = IF = DI = 2

units
Similarly, in

Δ

TKN and

Δ

RQU,

TK = KN = TH = 2

units

RQ = UQ = UR = 2

units
Also,

NC = RP = 2

units

Δ

HIG is also an equilateral triangle where G is the midpoint of IF.

2 IG = IF

\Rightarrow IG = \frac{IF}{2}

\Rightarrow IG = \frac{2}{2}

\Rightarrow IG = 1

unit
∴

IG = GH = IH = 1

unit
Similarly, in

Δ

MLK and

Δ

XQS,

ML = MK = LK = 1

unit

SQ = XS = QX = 1

unit
Also,

LN = XR = 1

unit
Now,
Perimeter of the given figure = Sum of all outer sides of the given figure

= AD + DI + IH + HG + GF + FE + EB + BT + TK + UQ + QS + XS + XR + PR + PA

= 4 + 2 + 1 + 1 + 1 + 2 + 4 + 4 + 2 + 1 + 1 + 1 + 2 + 4 + 4

+ 2 + 1 + 1 + 1 + 2 + 4

= 45 cm

Hence, the perimeter of the given figure is 45 units.

In the given figure all triangles are equilateral and AB = 8 units. Other triangles have been formed by taking the mid points of the sides. Then the perimeter of the figure is ____.

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