SECTION F — Assertion and Reasoning Type Questions [1 Marks Each]
Directions: Each of the following questions consists of two statements, namely, Assertion (A) and Reason (R). Select the correct option from the choices given below:
- (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
- (c) Assertion (A) is true, but Reason (R) is false.
- (d) Assertion (A) is false, but Reason (R) is true.
-
Q1. Assertion (A): All congruent triangles are similar, but all similar triangles need not be congruent.
Reason (R): Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional. -
Q2. Assertion (A): In $\triangle ABC$, if $DE \parallel BC$ intersecting $AB$ at $D$ and $AC$ at $E$, then $\frac{AD}{DB} = \frac{AE}{EC}$.
Reason (R): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. -
Q3. Assertion (A): If in two triangles $\triangle ABC$ and $\triangle PQR$, $\frac{AB}{QR} = \frac{BC}{PR} = \frac{CA}{PQ}$, then $\triangle ABC \sim \triangle QRP$.
Reason (R): Similarity of triangles is established under SSS criterion considering the correct vertex-to-vertex correspondence. -
Q4. Assertion (A): Diagonals of a trapezium $ABCD$ with $AB \parallel DC$ intersect each other at $O$, then $\frac{AO}{OC} = \frac{BO}{OD}$.
Reason (R): Any line parallel to the parallel sides of a trapezium divides the non-parallel sides proportionally. -
Q5. Assertion (A): The ratio of the areas of two similar triangles is equal to the square ofkes the square of the ratio of their corresponding altitudes.
Reason (R): The ratio of the areas of two similar triangles is equal to the ratio of their corresponding sides. -
Q6. Assertion (A): If sides of a triangle are $6\text{ cm}$, $8\text{ cm}$, and $10\text{ cm}$, it forms a right-angled triangle.
Reason (R): In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Converse of Pythagoras Theorem). -
Q7. Assertion (A): If $\triangle ABC \sim \triangle PQR$ such that $\text{area}(\triangle ABC) = 4 \text{ area}(\triangle PQR)$, then the ratio of their corresponding sides $\frac{AB}{PQ} = 2$.
Reason (R): The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. -
Q8. Assertion (A): Two poles of height $6\text{ m}$ and $11\text{ m}$ stand vertically on a plane ground. If the distance between their feet is $12\text{ m}$, the distance between their tops is $13\text{ m}$.
Reason (R): Pythagoras theorem can be applied to find the hypotenuse of the right triangle formed by the horizontal distance and height difference. -
Q9. Assertion (A): If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Reason (R): This statement represents the Converse of the Basic Proportionality Theorem. -
Q10. Assertion (A): In $\triangle ABC$, $\angle B = 90^\circ$ and $BD \perp AC$, then $\triangle DBA \sim \triangle DAB$ is false, but $\triangle DBA$ shares similarity with $\triangle CAB$.
Reason (R): If a perpendicular is drawn from the vertex containing the right angle of a right triangle to the hypotenuse, then triangles on both sides of the perpendicular are similar to the whole triangle and to each other. -
Q11. Assertion (A): The perimeters of two similar triangles $\triangle ABC$ and $\triangle PQR$ are $36\text{ cm}$ and $24\text{ cm}$. If $AB = 12\text{ cm}$, then the corresponding side $PQ = 8\text{ cm}$.
Reason (R): The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides. -
Q12. Assertion (A): In a rhombus $ABCD$, the sum of the squares of its sides is equal to the sum of the squares of its diagonals ($AB^2 + BC^2 + CD^2 + DA^2 = AC^2 + BD^2$).
Reason (R): The diagonals of a rhombus bisect each other at right angles ($90^\circ$), forming four congruent right-angled triangles. -
Q13. Assertion (A): If $\triangle ABC \sim \triangle DEF$ and $\angle A = 50^\circ$, $\angle E = 70^\circ$, then $\angle C = 60^\circ$.
Reason (R): Corresponding angles of similar triangles are equal, and the sum of angles in a triangle is $180^\circ$. -
Q14. Assertion (A): A vertical pole of length $20\text{ m}$ casts a shadow $10\text{ m}$ long on the ground. At the same time, a tower casts a shadow $50\text{ m}$ long, so the height of the tower is $100\text{ m}$.
Reason (R): At the same time of day, sun rays make equal angles with the ground, creating similar triangles between objects and their shadows. -
Q15. Assertion (A): If $D$ is a point on the side $BC$ of an equilateral triangle $ABC$ such that $BD = \frac{1}{3}BC$, then $9AD^2 = 7AB^2$.
Reason (R): Pythagoras theorem helps relate side lengths and altitude segments in equilateral triangles. -
Q16. Assertion (A): Two polygons with the same number of sides are similar if their corresponding angles are equal.
Reason (R): Equality of corresponding angles alone is not sufficient for polygon similarity; their corresponding sides must also be in the same ratio (proportional). -
Q17. Assertion (A): In $\triangle PQR$, if $PQ = 6\sqrt{3}\text{ cm}$, $PR = 12\text{ cm}$, and $QR = 6\text{ cm}$, then $\angle Q = 90^\circ$.
Reason (R): In a triangle, if the square of one side equals the sum of the squares of the other two sides, the angle opposite the first side is a right angle. -
Q18. Assertion (A): If the areas of two similar triangles are equal, the triangles are congruent.
Reason (R): When $\frac{\text{area}(1)}{\text{area}(2)} = 1$, the ratio of corresponding sides is $1$, meaning sides are equal. -
Q19. Assertion (A): If $ABC$ is an isosceles triangle with $AC = BC$ and $AB^2 = 2AC^2$, then $\triangle ABC$ is a right-angled triangle.
Reason (R): The condition satisfies the converse of Pythagoras theorem where the angle opposite to side $AB$ is $90^\circ$. -
Q20. Assertion (A): The line segments joining the mid-points of the sides of a triangle form four triangles, each of which is similar to the original triangle.
Reason (R): Mid-segment theorem states that the segment joining mid-points is parallel to the third side and half of it, creating proportional side scaling (SSS or SAS similarity).
Academic Repository Disclaimer & எச்சரிக்கை அறிவிப்பு :
English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
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