In the figure, AP and BQ equal and parallel are lines drawn at the ends of the line AB. The point of intersection of PQ and AB is marked as M.

i) Are the sides of Δ AMP equal to the sides of Δ BMQ? Why?

ii) What is special about the position of M on AB?

iii) Draw a line 5.5 centimeters long. Using a set square, locate the midpoint of this line.

Given-

AP = BQ

And AP ∥ BQ

According to the property,

If a pair of equal and opposite sides is parallel, then the four points connected form a parallelogram.

∴ the imaginary quadrilateral APBQ is a parallelogram.

AB and PQ are diagonals of parallelogram.

Also, the diagonals of parallelogram bisect each other.

∴ PM = MQ

And AM = MB

Also, ∠AMP = ∠BMQ, (vertically opposite angles)

According to property,

When two sides of a triangle and angle made by them are equal to the two sides and angle made by them of another triangle, then the third sides and the corresponding two angles are also equal.

∴ AP = BQ

Hence, the sides of the two triangles are equal.

ii) Specialty of point M,

It divides both sides in equal ratio i.e. 1:1

(Reason, Discussed above i.e. the imaginary quadrilateral APBQ is a parallelogram and AP,BQ are its diagonals).

iii) Steps for construction,

1. Draw a line AB = 5.5 cm

2. Using set-square draw two perpendicular lines on AB at each points A and B of length 5.5 cm as AP( = 5.5 cm) and BQ( = 5.5 cm)

3.Join lines PB and AQ and let them intersect at point O.

4.Drop a perpendicular from O to AB and let the new point be M.

M is the required midpoint of AB.