Question

The length and width of rectangle $ABCD$ are $10$ inches and $5$ inches, respectively. The rectangle is rotated ${45}^{\circ }$ in the clockwise direction to obtain rectangle $EFGH$.


Which of the given statements is/are true about the rectangles $ABCD\text{and}EFGH?$

Statement 1: Ratio of side $BC$ to area of rectangle $ABCD$ is equal to the ratio of side $FG$ to the area of the rectangle $EFGH.$
Statement 2: Perimeter of rectangle $EFGH=30$ inches

• A. $\text{Both statement 1 and 2}$
• B. $\text{Neither statement 1 nor 2}$
• C. $\text{Statement 1}$
• D. $\text{Statement 2}$

Answer 1

The correct option is A $\text{Both statement 1 and 2}$

Given: Clockwise rotation by ${45}^{\circ }$ of $ABCD$ produces the image $EFGH.$

Since rotation is a rigid transformation and rigid transformation produces congruent figures; therefore, $EFGH$ is congruent to $ABCD.$


$\Rightarrow$ Corresponding sides of rectangles $ABCD\text{and}EFGH$ are equal.
$\Rightarrow AB=EF=10\text{inches};BC=FG=5\text{inches};CD=GH=10\text{inches and}DA=HE=5\text{inches}$

$So,\text{Area of ABCD}=\text{Area of EFGH}=\text{Length}\times Width$
$=10\times 5$
$=50i{n}^2$

$\therefore$ Ratio of side $BC$ to area of rectangle $ABCD=\frac{5}{50}=\frac{1}{10}$

And, ratio of side $FG$ to the area of the rectangle $EFGH=\frac{5}{50}=\frac{1}{10}$

I.e., Ratio of side $BC$ to the area of rectangle $ABCD$ = Ratio of side $FG$ to the area of rectangle $EFGH$.
This means that statement $1$ is correct.

Also, since $EFGH\text{and}ABCD$ are congruent to each other
$\Rightarrow$ Perimeter of $EFGH$= Perimeter of $ABCD$ (congruent figures have same area and perimeter)

Now, perimeter of $ABCD=2\text{(Length + Width)}$(Perimeter of rectangle =$2\text{(Length + Width)})$

$\Rightarrow \text{Perimeter of ABCD}=2(10+5)=2\times 15=30\text{inches}$

$\Rightarrow \text{Perimeter of EFGH}=\text{Perimeter of ABCD = 30 inches}$

$\Rightarrow$$\text{Perimeter of EFGH = 30 inches}$
This means that statement $2$ is also correct.

➡Option C is correct.