Now, from point ‘S’ draws a line SQ perpendicular to the X-axis. We can find the TR received by the seller from the sale of OS unit of the commodity in two ways.

TR = Price x Quantity sold

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= OP x QQ

= Area OP SQ

Also, TR = Total of MR of all the OS units sold = ?MR

= Area OB AQ

By definition, TR should be the same by the two methods. Equating (2.3) and (2.4), we get

Area OPSQ = Area OBAQ

From the figure, we can see that

Area (OPSQ) = Area (OPRAQ + ASR)

Area (OBAQ) = Area (OPRAQ + BPR)

From equations (2.5), (2.6) and (2.7), we notice in terms of area that

OPRAQ + ASR = OPRAQ + BPR

Thus,

Area (ASR) = Area (BPR)

This implies that the areas of the triangles ASR and BPR are equal.

Further, In ?ASR and ?BPR

BPR = ASR (right-angles)

BRP = SRA (vertically opposite angles)

PBR = SAR (alternate angles)

Because all the three angles of the two triangles are equal to each other, the two triangles ASR and BPR are similar.

It is proved that both the triangles are equal in area and similar too. Thus, they must be congruent. If two triangles are congruent, their corresponding sides are equal. It means that

PR = RS

Hence, if AR and MR curves are straight lines, then, MR curve bisects the distance between the Y-axis and the AR curve. This relationship between the AR and MR curves can be used in drawing the two curves. But, this relationship between AR and MR curves is not true, if the two curves are either convex or concave to the origin.

As shown in Fig. 2.22 (a), if, AR and MR curves are convex to the origin, MR cuts the distance between the Y-axis and AR at less than half-way as measured from the Y-axis (i.e., to the left of the middle point). And, if, the AR and MR curves are concave to the origin, MR curve cuts the distance between the Y-axis and AR at more than half- way as measured from the Y-axis (see Fig. 2.22 (b)) in either of the two cases, MR curve will not lie half way from the Y-axis. The proof of this is beyond the scope of this book. 