Q3:

State whether the following quadrilaterals are similar or not:

Given: Two quadrilaterals, $PQRS$ and $ABCD$.

In quadrilateral $PQRS$: $PQ = 1.5\text{ cm}$, $QR = 1.5\text{ cm}$, $RS = 1.5\text{ cm}$, $SP = 1.5\text{ cm}$. All angles are $90^\circ$.

In quadrilateral $ABCD$: $AB = 3\text{ cm}$, $BC = 3\text{ cm}$, $CD = 3\text{ cm}$, $DA = 3\text{ cm}$. All angles are not $90^\circ$ (they are rhombic angles).

To Find: Determine whether the quadrilaterals $PQRS$ and $ABCD$ are similar.

Step 1: Define the conditions for similarity of polygons.

Two polygons of the same number of sides are similar if:

(i) Their corresponding angles are equal.

(ii) Their corresponding sides are in the same ratio (proportion).

Step 2: Check the ratio of corresponding sides.

Let us calculate the ratio of the sides of $PQRS$ to $ABCD$:

$\frac{PQ}{AB} = \frac{1.5}{3} = \frac{1}{2}$

$\frac{QR}{BC} = \frac{1.5}{3} = \frac{1}{2}$

$\frac{RS}{CD} = \frac{1.5}{3} = \frac{1}{2}$

$\frac{SP}{DA} = \frac{1.5}{3} = \frac{1}{2}$

[Since all ratios are equal to $\frac{1}{2}$, the sides are in proportion.]

Step 3: Check the corresponding angles.

In quadrilateral $PQRS$, all angles are $90^\circ$ (it is a square).

In quadrilateral $ABCD$, the angles are not $90^\circ$ (it is a rhombus).

Since the corresponding angles are not equal ($\angle P \neq \angle A$, $\angle Q \neq \angle B$, etc.), the condition for similarity is not satisfied.

Step 4: Conclusion.

For two polygons to be similar, both conditions (proportional sides and equal angles) must be satisfied simultaneously. Since the angles of $PQRS$ and $ABCD$ are not equal, the quadrilaterals are not similar.

Final Answer: The quadrilaterals $PQRS$ and $ABCD$ are not similar because, although their corresponding sides are in the same ratio, their corresponding angles are not equal.