CBSE Class 9 Mathematics Heron's Formula MCQs Set 23

Answer: (d) becomes four times

Explanation:
1. Doubling every side gives a similar triangle with scale factor 2.
2. Areas of similar figures scale as the square of the factor: 2² = 4.

Teacher's Note:
1. Area scales as the square of the side.
2. Check with Heron's formula: every term doubles.

Answer: (c) 3 : 1

Explanation:
1. Garden area = 4 × bed area.
2. Path area = 4 − 1 = 3 parts, bed = 1 part, so the ratio is 3 : 1.

Teacher's Note:
1. Similar triangles: area ratio is the square of the side ratio.
2. Path = garden − bed.

Answer: (d) 216 cm², 24 cm

Explanation:
1. s = 36 cm. Area = \(\sqrt{36 \times 18 \times 12 \times 6} = \sqrt{46656} = 216\) cm².
2. Smallest side = 18 cm, so altitude = 2 × 216 ÷ 18 = 24 cm.

Teacher's Note:
1. 18, 24, 30 is a right triangle, so area = (1/2) × 18 × 24 = 216.
2. Altitude = 2 × area ÷ base.

Answer: (a) 120 cm², 24 cm

Explanation:
1. s = 30 cm. Area = \(\sqrt{30 \times 20 \times 6 \times 4} = \sqrt{14400} = 120\) cm².
2. The longest altitude stands on the shortest side (10 cm): 2 × 120 ÷ 10 = 24 cm.

Teacher's Note:
1. Altitude is largest on the smallest side.
2. The triangle is right angled (10² + 24² = 26²).

Answer: (d) 2772 m², 36 m

Explanation:
1. Third side = 324 − 85 − 154 = 85 m. Then s = 162 m.
2. Area = \(\sqrt{162 \times 77 \times 77 \times 8} = 77 \times 36 = 2772\) m².
3. Perpendicular on 154 m = 2 × 2772 ÷ 154 = 36 m.

Teacher's Note:
1. Find the third side from the perimeter.
2. Use Heron's formula, then area = (1/2) base × height.

Answer: (a) Rs. 1219.68

Explanation:
1. s = 231 m. Area = \(\sqrt{231 \times 66 \times 88 \times 77} = 10164\) m².
2. Cost = 10164 × 0.12 = Rs. 1219.68.

Teacher's Note:
1. Convert paise to rupees (12 paise = Rs. 0.12).
2. Heron's formula gives the area first.

Answer: (b) 162 cm

Explanation:
1. Height = 2 × 360 ÷ 80 = 9 cm.
2. Equal side = \(\sqrt{40^2 + 9^2} = 41\) cm.
3. Perimeter = 80 + 2 × 41 = 162 cm.

Teacher's Note:
1. Drop the altitude to the base.
2. It bisects the base into 40 cm halves.

Answer: (d) 60 cm²

Explanation:
1. Legs add to 40 − 17 = 23 cm, and \(a^2 + b^2 = 289\).
2. \(2ab = 23^2 - 289 = 240\), so ab = 120.
3. The legs are 8 cm and 15 cm, so area = (1/2) × 120 = 60 cm².

Teacher's Note:
1. Use (a + b)² = a² + b² + 2ab.
2. 8, 15, 17 is a Pythagorean triple.

Answer: (b) 50 cm², 34.14 cm

Explanation:
1. Area = (1/2) × 10 × 10 = 50 cm².
2. Hypotenuse = \(10\sqrt2 = 14.14\) cm, so perimeter = 20 + 14.14 = 34.14 cm.

Teacher's Note:
1. Hypotenuse of a 45-45-90 triangle is leg × √2.
2. Perimeter includes all three sides.

Answer: (d) 6 minutes

Explanation:
1. 8 hectares = 80000 m², so the diagonal = \(\sqrt{2 \times 80000} = 400\) m.
2. Time = 0.4 km ÷ 4 km/h = 0.1 h = 6 minutes.

Teacher's Note:
1. 1 hectare = 10000 m².
2. Diagonal² = 2 × area.

Answer: (c) 48 m²

Explanation:
1. Half the longer diagonal is 24 m, so half the other diagonal is \(\sqrt{30^2 - 24^2} = 18\) m, and the other diagonal is 36 m.
2. Area = (1/2) × 48 × 36 = 864 m².
3. Each cow gets 864 ÷ 18 = 48 m².

Teacher's Note:
1. Diagonals of a rhombus cut at right angles.
2. Divide the total area by 18.