CBSE Class 9 Mathematics Constructions MCQs Set 06

Answer: (a) Rectangle

Explanation:
1. Opposite angles of a cyclic quadrilateral add up to 180°.
2. Opposite angles of a parallelogram are equal, so each is 90°.
3. A parallelogram with right angles is a rectangle.

Teacher's Note:
1. Combine the two angle properties.

Answer: (b) equal

Explanation:
1. The line of centres is the perpendicular bisector of the common chord.
2. Each point of intersection is joined to the two centres by equal radii in congruent triangles.
3. So the angles subtended at the two points of intersection are equal.

Teacher's Note:
1. The two points of intersection are mirror images in the line of centres.

Answer: (a) diagonals

Explanation:
1. The diagonals of a rhombus meet at right angles at O.
2. The angle in a semicircle is a right angle, so O lies on the circle with the side as diameter.

Teacher's Note:
1. Perpendicular diagonals give a right angle at their meeting point.

Answer: (a) Rectangle

Explanation:
1. Two chords bisect each other only if both pass through the centre, so they are diameters.
2. ABCD has equal diagonals that bisect each other, so it is a rectangle.

Teacher's Note:
1. Chords that bisect each other are diameters.

Answer: (c) Six

Explanation:
1. A triangle has 3 sides and 3 angles.
2. So it has 6 constituents.

Teacher's Note:
1. A triangle can be constructed when suitable three of its six constituents are given.

Answer: (a) Yes, the other side is about 7.04 cm

Explanation:
1. The hypotenuse is longer than the given side, so the triangle exists.
2. The other side = \( \sqrt{8.2^2 − 4.2^2} = \sqrt{49.6} \approx 7.04 \) cm.

Teacher's Note:
1. A right triangle exists if the given side is shorter than the hypotenuse.

Answer: (b) No, infinitely many similar triangles exist

Explanation:
1. Three angles fix the shape but not the size of the triangle.
2. Any enlargement or reduction has the same angles, so the triangle is not unique.

Teacher's Note:
1. At least one side is needed for a unique triangle.

Answer: (a) 15°

Explanation:
1. A 60° angle is constructed and bisected to get 30°, and bisecting again gives 15°.
2. The other angles cannot be obtained by repeated bisection of 60° or 90°.

Teacher's Note:
1. Angles 60°, 90° and their repeated bisections are constructible.

Answer: (a) 3 cm, 7.2 cm, 7.8 cm

Explanation:
1. Multiply each side by 3/5.
2. 5 × 0.6 = 3 cm, 12 × 0.6 = 7.2 cm, 13 × 0.6 = 7.8 cm.

Teacher's Note:
1. The scale factor multiplies every side.

Answer: (a) 5 cm, \( \frac{14}{3} \) cm, \( \frac{13}{3} \) cm

Explanation:
1. Multiply by 2/3: 7.5 × 2/3 = 5 cm, 7 × 2/3 = 14/3 cm, 6.5 × 2/3 = 13/3 cm.

Teacher's Note:
1. Scale every side by the same factor.

Answer: (a) 13 cm

Explanation:
1. Let the other side be x, so the hypotenuse is 18 − x.
2. (18 − x)² = x² + 12², so 324 − 36x = 144 and x = 5.
3. Hypotenuse = 13 cm.

Teacher's Note:
1. Use Pythagoras with the unknown side.

Answer: (b) No, because AC would have to be at least about 7.07 cm

Explanation:
1. For a given sum AB + BC = 10, the least possible AC occurs when AB = BC = 5, giving AC = \( 5\sqrt{2} \approx 7.07 \) cm.
2. AC = 7 cm is smaller than that, so no such triangle exists.

Teacher's Note:
1. Compare the given hypotenuse with the smallest possible value.

Answer: (b) No, because the two smaller sides only add up to the third

Explanation:
1. The sides would be 2 cm, 4 cm and 6 cm.
2. 2 + 4 = 6, which is not greater than 6.
3. The triangle inequality fails, so the triangle is degenerate and cannot be constructed.

Teacher's Note:
1. The sum of any two sides must exceed the third.

Answer: (a) 1.62 cm

Explanation:
1. For an equilateral triangle of side a, the inradius is \( \frac{a}{2\sqrt{3}} \).
2. r = \( \frac{5.6}{2 \times 1.732} \approx 1.62 \) cm.

Teacher's Note:
1. The incentre is also the centroid.

Answer: (a) \( 2\sqrt{13} \) cm

Explanation:
1. Half the base = \( \sqrt{7^2 − 6^2} = \sqrt{13} \) cm.
2. Base = \( 2\sqrt{13} \) ≈ 7.21 cm.

Teacher's Note:
1. The height bisects the base of an isosceles triangle.

Answer: (a) \( 2\sqrt{2} \) cm

Explanation:
1. Let each leg be a. Then 2a² = 16, so a = \( 2\sqrt{2} \) cm.

Teacher's Note:
1. Use Pythagoras with equal legs.