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Q3:
State whether the following quadrilaterals are similar or not:
State whether the following quadrilaterals are similar or not:
Solution :
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.
More Questions from Class 10 Mathematics Triangles EXERCISE 6.1
- Q1(i): Fill in the blanks using the correct word given in brackets : (i) All circles are ______. (congruent, similar)
- Q1(ii): Fill in the blanks using the correct word given in brackets : (ii) All squares are ______. (similar, congruent)
- Q1(iii): Fill in the blanks using the correct word given in brackets : (iii) All ______ triangles are similar. (isosceles, equilateral)
- Q1(iv): Fill in the blanks using the correct word given in brackets : (iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are ______ and (b) their corresponding sides are ______. (equal, proportional)
- Q2(i): Give two different examples of pair of (i) similar figures.
- Q2(ii): Give two different examples of pair of (ii) non-similar figures.
CBSE Solutions for Class 10 Mathematics Triangles
Chapters in CBSE - Class 10 Mathematics
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