In the given figure, PQRS is a square of side 2 cm and PLMN is a rectangle.

• The corner L of the rectangle lies on side QR.
• Side MN of the rectangle passes through the corner S of the square.

What is the area (in cm²) of the rectangle PLMN?

Note: The figure shown is representative

                          

The correct answer is 4.

Step-by-step Explanation:

Let us analyze the given figure containing the square PQRS of side length 2 cm and the rectangle PLMN. The corner L of the rectangle lies on the side QR of the square, and the side MN of the rectangle passes through the corner S of the square.

We can solve this problem using trigonometry by defining the orientation of the rectangle relative to the square.

Let the angle between the vertical side PQ of the square and the side PL of the rectangle be θ:

$\angle LPQ=\theta$

In the right-angled triangle ΔPQL (since ∠PQL = 90°):

$\cos \theta =\frac{PQ}{PL}$

Since PQ = 2 cm is the side of the square, the length PL of the rectangle is:

$PL=\frac{2}{\cos \theta }$

Now, let us determine the width PN of the rectangle. Since PLMN is a rectangle, the opposite side MN is parallel to PL. The distance between these two parallel lines is equal to the width of the rectangle, PN. Since the corner S of the square lies on the line MN, the perpendicular distance from S to the line PL is also equal to PN.

We can find this distance by projecting the segment PS onto the direction perpendicular to PL:

1. The side PS of the square is perpendicular to PQ, meaning ∠QPS = 90°.
2. Since ∠LPQ = θ, the angle between PL and PS is:

$\angle LPS=90°-\theta$

3. The width PN of the rectangle is the perpendicular distance from S to the line PL, which is given by:

$PN=PS\times \sin (\angle LPS)$

$PN=PS\times \sin (90°-\theta )=PS\times \cos \theta$

Since PS = 2 cm (the side of the square), we get:

$PN=2\cos \theta$

Now, we calculate the area of the rectangle PLMN:

$\text{Area}=PL\times PN$

Substituting the expressions for PL and PN:

$\text{Area}=\left(\frac{2}{\cos \theta }\right)\times \left(2\cos \theta \right)=4{\text{cm}}^2$

Thus, the area of the rectangle PLMN is always 4 cm².