This question was uploaded on 23/09/21 on social media accounts.
One line is drawn through 3 similar equilateral triangles such that the line starts from one edge of the first triangle and ends on the edge of the third triangle. Find the ratio of the intercepts made by the line with the tringles.
Solution 1:
\[\triangle OAB_1 \sim \triangle OAB_2 \sim \triangle OAB_3\]
Also,
\[OB_1 = OB_2 = OB_3\]
\[\Rightarrow OA_1 = A_1A_2 = A_2A_3\]
\[\Rightarrow a = x+b = y\]
Now,
\[\triangle OCB_1 \sim \triangle OA_3B_2
Also,
\[OB_1 = OB_2\]
\[\Rightarrow OC = CA_3\]
\[\Rightarrow a+x = b+y\]
By the above two conditions,
x = b and y = a
But a = x + b
\[\Rightarrow a = 2b\]
\[\Rightarrow a:b = 2:1\]
Solution 2:
Due to similar triangles and similar rhombus,
a = 2b
or
a:b = 2:1
Puzzle related to Geometry, Triangle, Length
