How Any Straight Line through the Center of a Rectangle Splits It into Two Equal Areas

The geometric properties of a rectangle are fascinating, especially when it comes to the relationship between a straight line passing through its center and the areas it creates. In this article, we will explore a geometric proof demonstrating that any straight line passing through the center of a rectangle splits it into two equal areas. This exploration will delve into the underlying principles and offer a detailed proof using both the length and width dimensions of the rectangle.

Definition and Initial Assumptions

Consider a rectangle PQRS with center G. A straight line VW passes through G, creating two trapeziums: PWVS and WQRV.

Similar Triangles and Congruence

First, we note that the two triangles GWY and VXG are similar. This similarity can be established because:

VX WY XG GY VG GW

These equalities imply that triangle GWY and triangle VXG are congruent. Therefore, the areas of these triangles are equal.

Calculating the Areas of Trapeziums

The area of the trapezium PWVS can be expressed as:

[ text{Area of trapezium} , PWVS text{Area of} , PYXS - text{Area of} , triangle GWY ]

Similarly, the area of trapezium WQRV can be computed as:

[ text{Area of trapezium} , WQRV text{Area of} , QYXR - text{Area of} , triangle GWY ]

Since the area of triangle GWY is equal to the area of triangle VXG, we have:

[ text{Area of trapezium} , PWVS text{Area of} , PYXS - text{Area of} , triangle GWY frac{1}{2} text{area of rectangle} , PQRS ]

And:

[ text{Area of trapezium} , WQRV text{Area of} , QYXR - text{Area of} , triangle GWY frac{1}{2} text{area of rectangle} , PQRS ]

Thus, the line through the center of the rectangle splits it into two equal areas.

Visual Proof Using Length and Width Dimensions

We illustrate this proof using two scenarios. In both scenarios, the original rectangle ABCD has dimensions AB CD L and BC AD W.

Scenario 1: Line Intersecting the Lengths of the Rectangle (Figure 1)

Consider a line PQ through O, the center of the rectangle, intersecting AB and CD at points P and Q, respectively. MN is a line through O parallel to AD and BC, making right-angled triangles ONP and OQM congruent. Since QM NP, and DM NB implies DQ PB, the areas of trapeziums APQD and PBCQ are equal.

The area of trapezium APQD is:

[ text{Area of trapezium} , APQD frac{1}{2}DQ times AB frac{1}{2}DQ times L times W ]

The area of trapezium OBCQ is:

[ text{Area of trapezium} , OBCQ frac{1}{2}PBCD times BC frac{1}{2}PBL times W ]

Since DQ PB, the areas of both trapeziums are equal, confirming that any line through the center of the rectangle divides it into two equal areas.

Scenario 2: Line Intersecting the Widths of the Rectangle (Figure 2)

Similar reasoning applies when the line intersects the widths of the rectangle. The areas of trapeziums CDPQ and ABQO are proven to be equal, confirming the same property as in the previous scenario.

Conclusion and Algebraic Implication

The original area of a rectangle A is given by:

[ A l times w ]

When any straight line through the center splits the rectangle into two trapeziums, the new areas NA are either:

[ NA frac{l}{2} times w ] or NA l times frac{w}{2}

In both cases, the new area is:

[ NA l times frac{w}{2} ]

This is half of the original area of A l times w. This is a simple algebraic implication that dividing something in half results in half the area.

Key Takeaways

Any straight line through the center of a rectangle divides it into two equal areas. The geometric proof involves congruent triangles and the properties of trapeziums. Algebraic implications confirm that splitting a rectangle along its center line results in areas of half the original size.

Understanding these principles provides insight into the fundamental properties of rectangles and their symmetry, making it easier to visualize and apply these geometric concepts in various real-world scenarios.