Segment drawn on scale having end points O (0,0) and A [tex](\frac{3}{4},\frac{9}{10})[/tex] is a line segment.
O A= [tex]\sqrt{[\frac{3}{4} -0]^{2} +[\frac{9}{10} -0]^{2}\\\\[/tex]
O A = [tex]= \sqrt{\frac{9}{16}+\frac{81}{100}[/tex]
= [tex]\sqrt\frac{549}{400}[/tex]
= [tex]\frac{\sqrt{549}}{20}[/tex]
Now , the same segment actual structure having end points O'(0,0) and B (30,36) is also a line segment.
O'B= [tex]\sqrt{(30-0)^{2}+(36-0)^{2}[/tex]
= [tex]\sqrt{900+1296}[/tex]
= [tex]\sqrt {2196}[/tex]
= 2√549
[tex]\frac{\text{Actual length}}{\text{Length on scale}}=\frac{OB'}{OA}=\frac{2\sqrt549}{\frac{\sqrt549}{20}}=40[/tex] [Cancelling √549 from numerator and denominator]
So, Actual length = 40 × Length on scale
