Question 1:

Draw a line segment of length 7.6 cm and divide it in the ratio 5:8. Measure the twoparts. Give the justification of the construction.

Answer:

A line segment of length 7.6 cm can be divided in the ratio of 5:8 as follows.

Step 1 Draw line segment AB of 7.6 cm and draw a ray AX making an acute angle with line segment AB.

Step 2 Locate 13 (= 5 + 8) points, A₁, A₂, A₃, A₄ …….. A₁₃, on AX such that AA₁ = A₁A₂ = A₂A₃ and so on.

Step 3 Join BA₁₃.

Step 4 Through the point A5, draw a line parallel to BA₁₃ (by making an angle equal to ∠AA₁₃B) at A₅ intersecting AB at point C.

C is the point dividing line segment AB of 7.6 cm in the required ratio of 5:8.

The lengths of AC and CB can be measured. It comes out to 2.9 cm and 4.7 cm respectively.

Justification

The construction can be justified by proving thata

AC/CB = 5/8

By construction, we have A₅C || A₁₃B. By applying Basic proportionality theorem for the triangle AA₁₃B, we obtain

AC/CB = AA₅/A₅A₁₃ ….   (1)

From the figure, it can be observed that AA₅ and A₅A₁₃ contain 5 and 8 equal divisions of line segments respectively.

∴ AA₅/A₅A_{13 = 5/8 …..  (2)}

On comparing equations (1) and (2), we obtain

AC/CB = 5/8

This justifies the construction.

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