Question

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


Which of the given statements about rectangles $ABCD$ and $EFGH$ are true?

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

• A. Neither statement $1$ nor $2$
• B. Statement $1$
• C. Statement $2$
• D. Both statements $1$ and $2$

Answer 1

The correct option is D Both statements $1$ and $2$

Given: Clockwise rotation of $ABCD$ by $45°$ produces the image $EFGH$.
Since rotation is a rigid transformation and rigid transformation produces congruent figures, $EFGH$ is congruent to $ABCD$.

Corresponding sides of rectangles $ABCD$ and $EFGH$ are equal.
$\Rightarrow AB=EF,BC=FG,CD=GH,andDA=HE$

Also, congruent figures have the same area and perimeter, i.e.,
Area of $EFGH=$ area of $ABCD$, and
Perimeter of $EFGH=$ perimeter of $ABCD$

$\Rightarrow \frac{BC}{AreaofABCD}=\frac{FG}{AreaofABCD}$

Statement 1 is correct.

Perimeter of $EFGH=$ perimeter of $ABCD$
$=2$(length + width)
$=2(10+5)$
$=30$ in

Statement $2$ is also correct.