State true of false. Give reason also.

By geometrical construction, it is possible to divide a line segment in the ratio 3+root 2 : 3-root 2 .

Yes, it is possible to divide a line segment in the ratio by geometrical construction.

Steps of construction are given below.

Step 1: Draw a line segment AB.

Step 2: Draw any ray AD, making an acute angle ∠BAD with AB.

Step 3: Along AD, mark off point A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6} such that AA_{1} = A_{1}A_{2} = A_{2}A_{3} = A_{3}A_{4 }= A_{4}A_{5} = A_{5}A_{6 }= 1 units

Step 4: Join BA_{6.}

Step 5: Draw A_{4}C ⊥ AD at A_{4} such that A_{4}C = 1 unit.

Step 6 : Join A_{3}C.

Step 7: With A_{3 }as centre and radius = A_{3}C draw an arc intersecting AD in X.

Step 8: Draw XY || A_{6}B, intersecting AB in Y.

Here, Y divides AB in the ratio .

Justification:

ΔAXY ∼ ΔABA_{6 }(AA similarity)

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