In the given figure, all triangles are equilateral and $A B = 8$ units. Other triangles have been formed by taking the mid-points of the sides. What is the perimeter of the figure?

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Solution

Firstly, find the all outer sides of the given triangles and then find the perimeter by using sum of all sides of the triangles.

Given, $△ A B C$ is an equilateral triangle.

Here, $A B = 8$ units

$\Rightarrow A B = B C = C A = 8$ units.
Since, $△ A B C$ is an equilateral triangle, then the triangle formed by mid-points of the sides is also an equilateral triangle.

Thus, $△ A D E$ is an equilateral triangle.

Here, $E$ is the mid-point of $A B$ .

$\Rightarrow A E = B E = \frac{A B}{2} = \frac{8}{2} = 4$ units

Now, in $△ A D E$ ,

$A D = D E = E A = 4$ units

Similarly, $△ B O T$ and $△ U P C$ are equilateral triangles and having each sides equal.

i.e., $B O = O T = B T = U C = P C = P U = 4$ units

It is also clear that, $O C = P A = 4$ units

Also, $△ D I F$ is an equilateral triangle.

Here, $F$ is the mid-point of $D E$ .

$\Rightarrow D F = F E = \frac{D E}{2} = \frac{4}{2} = 2$ units

In $△ D I F, D I = I F = D F = 2$ units

Similarly, in $△ T K N$ and $△ R Q U$ ,

$T K = K N = T N = R Q = U Q = U R = 2$ units

it is also clear that, $N C = R P = 2$ units

Also, $△ H I G$ is an equilateral triangle.

Here, $G$ is the mid-point of $I F$ .

$\Rightarrow I G = G F = \frac{I F}{2} = \frac{2}{2} = 1$ unit

Similarly, in $△ M L K$ and $△ X Q S$ ,

$M L = M K = L K = S Q = X S = Q X = 1$ unit

it is also clear that, $L N = X R = 1$ unit

Now, perimeter of the given figure

$=$ Sum of all outer sides of the given figure

$= A D + D I + I H + H G + G F + F E + E B + B T + T K + K M + L M + L N + N O + O C + C U + U Q + Q S + X S + X R + P R + P A$

$= [4 + 2 + 1 + 1 + 1 + 2 + 4 + 4 + 2 + 1 + 1 + 1 + 2 + 4 + 4 + 2 + 1 + 1 + 1 + 2 + 4] c m$

$= 45 c m$

Hence, the perimeter of the given figure is $45 c m$ .

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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. What is the perimeter of the figure?

In the given figure, all triangles are equilateral and $A B = 8$ units. Other triangles have been formed by taking the mid-points of the sides. What is the perimeter of the figure?