CBSE Class 9 Maths Ganita Manjari Part 2 Ch 12 Quadrilaterals MCQs with Answers Set 01

Answer: (c) two of its sides cross each other

Explanation:
1. The definition demands that every point of the four segments, apart from the vertices, lies on exactly one segment.
2. A crossing point lies on two segments at once, so the rule fails. Such a figure is called self-intersecting.
3. A reflex angle is allowed: it just gives a non-convex quadrilateral.

Teacher's Note:
1. Link the answer to the words 'exactly one'.
2. Non-convex is still a quadrilateral; self-intersecting is not.

Answer: (b) ABDC

Explanation:
1. A valid name must trace the boundary: start at any vertex and go round in either direction.
2. The eight names are ABCD, BCDA, CDAB, DABC and their reversals DCBA, ADCB, BADC, CBAD.
3. ABDC jumps from B to D, which is a diagonal, so it describes a different figure.

Teacher's Note:
1. Consecutive letters must be joined by a side.
2. Check each option letter pair by letter pair.

Answer: (a) every one of its internal angles is less than 180°

Explanation:
1. A convex quadrilateral has no dent.
2. A dent appears as an internal angle bigger than 180° (a reflex angle).
3. So convex means all four internal angles are below 180°. The other options describe only special convex shapes.

Teacher's Note:
1. Think 'no dent'.
2. Parallel sides and equal diagonals are extra properties, not the definition.

Answer: (d) do not intersect each other

Explanation:
1. In a convex quadrilateral the diagonals always cross inside the figure.
2. In a non-convex one the dent pushes one diagonal outside, so the two never meet.
3. This gives a second test for convexity.

Teacher's Note:
1. Draw a dart shape and its two diagonals.
2. Diagonal test: cross = convex; no meeting = non-convex.

Answer: (b) DC

Explanation:
1. Opposite sides share no vertex.
2. BC shares B with AB, and AD shares A, so both are adjacent.
3. DC shares neither A nor B, so it is opposite. AC is a diagonal, not a side.

Teacher's Note:
1. Adjacent = share a vertex; opposite = share none.
2. Rule out diagonals first.

Answer: (c) 85°

Explanation:
1. The angles of a quadrilateral add up to 360°.
2. 80 + 95 + 100 = 275.
3. Fourth angle = 360° − 275° = 85°.

Teacher's Note:
1. The 360° comes from two triangles made by a diagonal.
2. Add carefully before subtracting.

Answer: (a) a rectangle

Explanation:
1. Adjacent angles of a parallelogram are co-interior, so they add to 180°.
2. If one angle is 90°, its neighbour is 180° − 90° = 90°, and opposite angles are equal too.
3. All four angles are right angles, so it is a rectangle. The sides need not be equal, so it need not be a square.

Teacher's Note:
1. One right angle forces all four.
2. Square needs equal sides as well.

Answer: (d) a rectangle

Explanation:
1. The diagonals of a parallelogram bisect each other.
2. If they are also equal, the congruent triangles formed force every angle to be 90°.
3. So it is a rectangle, but not always a square: a 3 × 4 rectangle has equal diagonals of 5.

Teacher's Note:
1. Equal diagonals give right angles, not equal sides.
2. Use the 3–4–5 rectangle as a quick example.

Answer: (b) false, because the diagonals of every rhombus are already perpendicular

Explanation:
1. Every rhombus already has perpendicular diagonals.
2. So the extra condition adds nothing and cannot turn a rhombus into a square.
3. Example: the rhombus (0, 0), (4, 3), (8, 0), (4, −3) has sides 5 and diagonals 8 and 6, so it is not a square.

Teacher's Note:
1. Ask: is the extra condition already true?
2. A condition that is always true narrows nothing.

Answer: (c) SSS

Explanation:
1. Only sides are given: two pairs of equal opposite sides.
2. The diagonal AC is common, which gives a third pair of equal sides.
3. So △ACD ≅ △CAB by SSS, and the equal angles then show the sides are parallel.

Teacher's Note:
1. Given only sides, use SSS.
2. This proof runs Theorem 1(a) backwards.

Answer: (a) ASA

Explanation:
1. The diagonal AC cuts both pairs of parallel sides, giving two pairs of equal alternate angles.
2. The common side AC lies between those angles.
3. So △ACD ≅ △CAB by ASA, and AB = DC, AD = BC follow.

Teacher's Note:
1. Parallel lines give angles, so expect ASA.
2. Name the included side AC.

Answer: (d) 7 cm

Explanation:
1. By the Midpoint Theorem, PQ is parallel to BC and half of it.
2. PQ = 14 ÷ 2 = 7 cm.
3. (a) doubles instead of halving; (c) halves twice.

Teacher's Note:
1. Half, not double.
2. The segment is also parallel to BC.

Answer: (b) bisects the third side

Explanation:
1. The theorem starts from two midpoints and concludes 'parallel and half'.
2. The converse starts from one midpoint and a parallel line.
3. It concludes that the line meets the third side at its midpoint, that is, it bisects it.

Teacher's Note:
1. Compare what is given and what is proved.
2. Halving the length also follows, but bisecting is the key conclusion.

Answer: (c) 2:1, with the longer part next to the vertex

Explanation:
1. The medians meet at the centroid.
2. The part from the vertex to the centroid is twice the part from the centroid to the side.
3. So the ratio is 2:1 with the longer part at the vertex; (d) reverses it.

Teacher's Note:
1. Direction matters in a ratio.
2. Vertex side = 2 parts, side = 1 part.

Answer: (a) 12 cm

Explanation:
1. The centroid divides the median 2:1 from the vertex.
2. CM = \( \frac{2}{3} \) × 18 = 12 cm.
3. The remaining piece MP = 6 cm is option (c).

Teacher's Note:
1. Vertex portion is two-thirds.
2. Check: 12 + 6 = 18.

Answer: (d) a parallelogram

Explanation:
1. This is Varignon's theorem (the Midpoint Theorem for Quadrilaterals).
2. Each diagonal cuts off two triangles; the Midpoint Theorem makes a pair of opposite sides parallel to that diagonal.
3. So both pairs of opposite sides are parallel. It is a rhombus, rectangle or square only in special cases.

Teacher's Note:
1. No condition on the original quadrilateral is needed.
2. 'Always' points to the most general answer.

Answer: (b) AC = BD

Explanation:
1. The sides of the Varignon parallelogram are ½AC and ½BD.
2. Adjacent sides are equal exactly when AC = BD.
3. A parallelogram with equal adjacent sides is a rhombus. Perpendicular diagonals give a rectangle instead.

Teacher's Note:
1. Sides come from diagonal lengths.
2. Angles come from the angle between the diagonals.

Answer: (c) AC and BD are perpendicular

Explanation:
1. One pair of sides of the Varignon parallelogram is parallel to AC, the other to BD.
2. So its angles equal the angle between AC and BD.
3. They are right angles exactly when AC ⊥ BD.

Teacher's Note:
1. Pair this with the previous question.
2. Lengths → rhombus; angle → rectangle.

Answer: (a) any quadrilateral whatsoever

Explanation:
1. The four angles of every quadrilateral add to 360°.
2. So four copies, one of each angle, fit exactly round a point.
3. The chapter's methods work for any quadrilateral, convex or not.

Teacher's Note:
1. The key fact is the 360° angle sum.
2. Even a non-convex quadrilateral tiles.

Answer: (d) its internal angle of 108° does not divide 360° a whole number of times

Explanation:
1. At each vertex of a tiling the angles must total 360°.
2. 360 ÷ 108 = 3⅓, not a whole number.
3. Three pentagons leave a 36° gap; four overlap. The pentagon is convex, so (a) is false.

Teacher's Note:
1. Test: does the angle divide 360 exactly?
2. Show the 36° gap.

Answer: (b) four congruent triangles

Explanation:
1. Each joining segment is half of the side it is parallel to.
2. All four small triangles have sides equal to half of each side of the original.
3. So all four are congruent by SSS, each a half-size copy.

Teacher's Note:
1. The middle triangle is also congruent.
2. Each small triangle is similar to the big one.

Answer: (c) 12 cm

Explanation:
1. This segment equals the average of the parallel sides.
2. (9 + 15) ÷ 2 = 12 cm.
3. (a) is the difference; (b) is half the difference.

Teacher's Note:
1. Average, so it lies between 9 and 15.
2. It is also parallel to both parallel sides.

Answer: (d) 20

Explanation:
1. From each vertex there are n − 3 diagonals.
2. Each diagonal is counted twice, so the total is \( \frac{n(n-3)}{2} \).
3. For n = 8: \( \frac{8 \times 5}{2} \) = 20.

Teacher's Note:
1. Remember to halve.
2. n − 3: not itself and not its two neighbours.

Answer: (a) 720°

Explanation:
1. An n-gon splits into n − 2 triangles from one vertex.
2. Angle sum = (n − 2) × 180°.
3. For n = 6: 4 × 180° = 720°.

Teacher's Note:
1. 540° is the pentagon, 900° the heptagon.
2. Draw the triangles once to remember.

Answer: (b) one of its internal angles is greater than 180°

Explanation:
1. Figure 2 has a dent. At that vertex the internal angle is more than a straight angle.
2. One reflex angle is exactly what makes a quadrilateral non-convex.
3. That is also why its dashed diagonals do not meet. Unequal sides or no parallel sides happen in convex shapes too.

Teacher's Note:
1. Point to the dented vertex.
2. Link the reflex angle with the diagonal test.

Answer: (c) 10 cm

Explanation:
1. Each side must be less than the sum of the other three.
2. d < 2 + 5 + 11 = 18, and 11 < 2 + 5 + d gives d > 4.
3. So 4 < d < 18. Only 10 cm fits.

Teacher's Note:
1. Check both the largest given side and d.
2. 4 cm fails because the inequality is strict.

Answer: (d) the reasoning 'if x = y then x² = y²' cannot be reversed

Explanation:
1. 'If x = y then x² = y²' is correct.
2. Reversing it fails: x = 3 and y = −3 have equal squares but are not equal.
3. So every backward step needs its own check. Option (a) states the false claim itself.

Teacher's Note:
1. Squaring loses the sign.
2. One counterexample kills a reversed step.

Answer: (a) If the diagonals of a quadrilateral are equal, then it is a rectangle.

Explanation:
1. (b), (c) and (d) are the chapter's true tests for a parallelogram.
2. Equal diagonals alone are not enough: an isosceles trapezium has equal diagonals and no right angle.
3. Equal diagonals make a parallelogram a rectangle, not any quadrilateral.

Teacher's Note:
1. The isosceles trapezium is the standard counterexample.
2. Watch for a dropped hypothesis.

Answer: (c) 2023

Explanation:
1. The 'hat' was found in 2023 by Smith, Myers, Kaplan and Goodman-Strauss.
2. It was the first single shape known to tile the plane aperiodically.
3. 2011 is the year of Dan Shechtman's Nobel Prize for quasicrystals, a different discovery.

Teacher's Note:
1. Aperiodic = never repeats.
2. Do not mix it up with the 2011 Nobel Prize.

Answer: (b) a square

Explanation:
1. Bisecting diagonals make it a parallelogram.
2. Perpendicular diagonals then make it a rhombus (all sides equal).
3. Equal diagonals make it a rectangle (all angles 90°). A rhombus that is a rectangle is a square.

Teacher's Note:
1. Add one condition at a time.
2. Square = rhombus + rectangle.