Ex.11.1 Q1 Constructions Solution - NCERT Maths Class 10

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Question

Draw a line segment of length \(7.6 \,\rm{cm}\) and divide it in the ratio \(5:8\). Measure the two parts.

 

Text Solution

 

What is known?

Length of line segment and the ratio to be divided.

What is unknown?

Construction

Reasoning:

  • Draw the line segment of given length.
  • Then draw another line which makes an acute angle with the given line.
  • Divide the line into \(m + n\) parts where \(m\) and \(n\) are the ratio given.
  • Basic proportionality theorem states that, “If a straight line is drawn parallel to a side of a triangle, then it divides the other two sides proportionally".

Steps:

(i) Draw \({AB = 7.6\, \rm{cm}}\)

(ii) Draw ray \(AX,\) making an acute angle width \(AB.\)

(iii) Mark \(13\;(=\;5\;+\;8)\) points \({A_1},\,{A_2},\,\ldots.\,{A_13}\) on \(AX\) such that\(\text{A}{{\text{A}}_{\text{1}}}\text{=}\,{{\text{A}}_{\text{1}}}{{\text{A}}_{\text{2}}}\text{=}{{\text{A}}_{2}}{{\text{A}}_{3}}\text{=}.....{{\text{A}}_{\text{12}}}{{\text{A}}_{\text{13}}}\)

(iv)  Join \({{BA}_{13}}\)

(v) Through \(A_5\) (since we need \(5\) parts to \(8\) parts) draw  \({{C}}{{{A}}_{{5}}}\) parallel to \({{B}}{{{A}}_{{{13}}}}\) where \(C\) lies on \(AB.\)

Now \({{AC: CB = 5:8}}\)

We find \(AC = 2.9 \,\rm{cm}\) and \(CB = 4.7 \,\rm{cm}\)

Proof:

\({{C}}{{{A}}_{{5}}}\) is parallel to \({{B}}{{{A}}_{{{13}}}}\)

By Basic Proportionality theorem, in \({{\Delta A}}{{{A}}_{{{13}}}}{{B}}\)

 \(\begin{align}\frac{{AC}}{{CB}} = \frac{{{{A}}{{{A}}_{{5}}}}}{{{{{A}}_{{5}}}{{{A}}_{{{13}}}}}}{{ = }}\frac{{{5}}}{{{8}}}\,\,\end{align}\) (By Construction)

Thus, \(C\) divides \(AB\) in the ratio \(5:8\).

  
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