This question was uploaded on 14/09/21 on social media accounts.
Three parallel lines with one end on the same line. The top of the first line and bottom of the second line is joined and vice versa to draw the second line. One more parallel line is drawn from the intersection of both lines to the initial line. Find the length of the third line if the length of the other two parallel lines is 5, 3.
Solutions:
Method 1:
For the general case, if lengths are a, b instead of 5, 3.
Then,
\[\frac{1}{x}=\frac{1}{a}+\frac{1}{b}\]
\[\Rightarrow x=\frac{a+b}{ab}\]
For this case,
\[\frac{1}{x}=\frac{1}{5}+\frac{1}{3}\]
\[\Rightarrow x=\frac{15}{8}\]
Method 2:
Triangles OAB and OCD are similar,
a, b, c are proportionality constants.
OA, OD and OB, OD are in the ratio 5:3 (AB:CD)
Now Triangles OMD and ABD are similar,
By this BM:MD = AO:OD = 5:3
Now,
AB:OM = AD:OD = BD:MD = 8:3
⇒ 5:x = 8:3
⇒ x = 15/8
Puzzle related to Geometry, Triangle
