A rectangle has a length L and a width W, where L > W. If the width, W, is increased by 10%, which one of the following statements is correct for all values of L and W?

Rectangle Geometry: Width Increase Analysis

This solution examines how a 10% increase in a rectangle's width affects its properties.

Given: Rectangle dimensions are length $L$ and width $W$, with $L>W$.

Modification: Width $W$ is increased by 10%.

New Width: $W^{\prime }=W+0.10W=1.1W$. The length $L$ remains constant.

Analyzing Potential Changes

- Perimeter: Original Perimeter $P=2(L+W)$. New Perimeter $P^{\prime }=2(L+W^{\prime })=2(L+1.1W)$. The percentage change is calculated as $\frac{P^{\prime }-P}{P}\times 100$. Substituting the values: $\frac{2(L+1.1W)-2(L+W)}{2(L+W)}\times 100=\frac{0.2W}{2(L+W)}\times 100=\frac{10W}{L+W}\%$. This value depends on the ratio $L:W$ and is not always 10%.
- Diagonal Length: Original Diagonal $D=\sqrt{L^2+W^2}$. New Diagonal $D^{\prime }=\sqrt{L^2+(W^{\prime }{)}^2}=\sqrt{L^2+(1.1W{)}^2}=\sqrt{L^2+1.21W^2}$. The percentage change is $\frac{D^{\prime }-D}{D}\times 100=\frac{\sqrt{L^2+1.21W^2}-\sqrt{L^2+W^2}}{\sqrt{L^2+W^2}}\times 100$. This value depends on the ratio $L:W$ and is not always 10%.
- Area: Original Area $A=L\times W$. New Area $A^{\prime }=L\times W^{\prime }=L\times (1.1W)=1.1(LW)$. Therefore, $A^{\prime }=1.1A$. The percentage change in area is $\frac{A^{\prime }-A}{A}\times 100=\frac{1.1A-A}{A}\times 100=\frac{0.1A}{A}\times 100=10\%$. This is true for all values of $L$ and $W$.
- Shape Transformation: A rectangle becomes a square when its length equals its width ($L=W^{\prime }$). In this case, it would require $L=1.1W$. This condition ($\frac{L}{W}=1.1$) is specific and does not apply to all rectangles where $L>W$. Thus, the rectangle does not necessarily become a square.

Correct Statement Identification

Based on the analysis, the only statement that remains correct for all possible values of $L$ and $W$ (given $L>W$) is that the area increases by 10% when the width is increased by 10%.