Class 9 Maths: Perimeter and Area – Extra Questions

Ans: D

Explanation: The diagonals of a parallelogram intersect at their midpoints. This creates four triangles. These four triangles are congruent (by Side-Angle-Side congruence). Since the diagonals bisect each other, and they are also the bisectors for the sides of the parallelogram. These four triangles also have equal area. Option A states that they are triangles of equal area, which is true. Option B states they are congruent triangles, which is also true. Thus, both are correct.

Correct Option: D

Ans: B

Explanation: The area of a parallelogram is given by the formula: Area = base * height. Since the two parallelograms share the same base length and are positioned between the same parallel lines, their heights must also be the same. Therefore, their areas are equal. The ratio of their areas is 1:1.

Correct Option: B

Ans: A

Solution:
Area of a rhombus = (1/2) * d1 * d2, where d1 and d2 are diagonals.
Area = (1/2) * 6 cm * 8 cm = 24 sq cm

Answer: 24

Ans: A

Solution: Area = (1/2) * base * height
Answer: 187.5 cm²

Ans: A

Solution:
Since $\triangle ABC \cong \triangle DEF$, the corresponding sides are equal. Therefore, $DE = AB = 5$, $EF = BC = 7$, and $DF = AC = 9$.
The perimeter of $\triangle DEF$ is $DE + EF + DF = 5 + 7 + 9 = 21$.
Answer: 21

Ans: A

Solution:
Nose cone area: (1/2) * 3 * 4 = 6 sq cm
Body area: 10 * 2 = 20 sq cm
Two fins area: 2 * (1/2) * 2 * 3 = 6 sq cm
Top fin area: 5 * 1 = 5 sq cm
Total area: 6 + 20 + 6 + 5 = 37 sq cm
Answer: 37 sq cm

Ans: A

Explanation: The area of a trapezoid is given by the formula A = (1/2) * (b1 + b2) * h, where A is the area, b1 and b2 are the lengths of the parallel sides (bases), and h is the height (depth in this case). We are given A = 640 square meters, b1 = 10 meters, and b2 = 6 meters. We need to solve for h.
640 = (1/2) * (10 + 6) * h
640 = (1/2) * 16 * h
640 = 8 * h
h = 640 / 8
h = 80

Correct Answer: 80

Ans: C

Explanation: The area of an equilateral triangle with side length $s$ is given by $A = \frac{s^2\sqrt{3}}{4}$.
We are given that $A = 36\sqrt{3}$.
So, $\frac{s^2\sqrt{3}}{4} = 36\sqrt{3}$.
Dividing both sides by $\sqrt{3}$, we get $\frac{s^2}{4} = 36$.
Multiplying both sides by 4, we get $s^2 = 144$.
Taking the square root of both sides, we get $s = 12$ cm (since side length must be positive).
The perimeter of the equilateral triangle is $3s = 3(12) = 36$ cm.

Correct Option: C

Ans: D

Solution:
Let the base be $b=20$ cm and the area be $A=120$ $cm^2$.
The height $h$ is found by $A = \frac{1}{2}bh$, so $120 = \frac{1}{2}(20)h$, which gives $h = 12$ cm.
Since the triangle is isosceles, the height bisects the base.
We have two right triangles with legs of length $10$ cm and $12$ cm.
The hypotenuse (side) is $s = \sqrt{10^2 + 12^2} = \sqrt{100+144} = \sqrt{244} = 2\sqrt{61}$ cm.
The perimeter is $2s + b = 2(2\sqrt{61}) + 20 = 20 + 4\sqrt{61}$ cm.
Answer: $20+4\sqrt{61}$ cm

Ans: D

Explanation: First, we recognize that the sides 18 cm, 24 cm, and 30 cm form a right triangle because they satisfy the Pythagorean theorem (18^2 + 24^2 = 324 + 576 = 900 = 30^2). Therefore, the area of the triangle can be calculated as (1/2) * base * height, where the base and height are the two shorter sides. Area = (1/2) * 18 cm * 24 cm = 216 cm^2. The shortest side is 18 cm. Let ‘h’ be the altitude to the shortest side. The area can also be expressed as (1/2) * 18 cm * h. So, 216 cm^2 = (1/2) * 18 cm * h. Solving for h: h = (2 * 216 cm^2) / 18 cm = 24 cm.

Correct Answer: Area = 216 cm^2; Altitude to shortest side = 24 cm

Ans: B

Explanation: The arc length (s) of a circle is related to the radius (r – which is the length of the pendulum) and the central angle (θ) in radians by the formula s = rθ. We are given the arc length (8.8 cm) and the angle in degrees (30 degrees). First, convert the angle to radians: 30 degrees * (π / 180 degrees) = π/6 radians. Now we can use the formula: 8.8 cm = r * (π/6). Solve for r: r = 8.8 cm * (6/π) ≈ 16.76 cm.

Correct Answer: 16.76 cm

Ans: D

Solution:
Let $s$ be the side length of the equilateral triangle. The area of an equilateral triangle is given by the formula $A = \frac{s^2\sqrt{3}}{4}$. We are given that the area is $16\sqrt{3}$ cm$^2$. Thus we have
$\frac{s^2\sqrt{3}}{4} = 16\sqrt{3}$
$s^2\sqrt{3} = 64\sqrt{3}$
$s^2 = 64$
$s = \sqrt{64}$
Since $s$ must be positive, $s=8$.
Answer: 8

Ans: B

Solution:
Use the Pythagorean theorem to find the other side: a^2 + b^2 = c^2. 8^2 + b^2 = 17^2, 64 + b^2 = 289, b^2 = 225, b = 15. Perimeter = 2(length + width) = 2(8 + 15) = 2(23) = 46.
Answer: 46 meters

Ans: C

Solution:
Let $s$ be the side length of the equilateral triangle.
The area of an equilateral triangle is given by the formula $A = \frac{s^2\sqrt{3}}{4}$.
We are given that the area is $25\sqrt{3}$ cm$^2$.
So, $\frac{s^2\sqrt{3}}{4} = 25\sqrt{3}$.
Multiplying both sides by 4 gives $s^2\sqrt{3} = 100\sqrt{3}$.
Dividing both sides by $\sqrt{3}$ gives $s^2 = 100$.
Taking the square root of both sides gives $s = 10$ cm.
The perimeter of the equilateral triangle is $3s = 3(10) = 30$ cm.
Answer: 30 cm

Ans: C

Explanation:
1. Identify the triangle type: The triangle has two equal sides (65m and 65m), making it an isosceles triangle.

2. Calculate the area: We can use Heron’s formula to find the area of the triangle.
* First, find the semi-perimeter (s): s = (a + b + c) / 2 = (50 + 65 + 65) / 2 = 180 / 2 = 90 meters.
* Then, apply Heron’s formula: Area = sqrt[s(s-a)(s-b)(s-c)] = sqrt[90(90-50)(90-65)(90-65)] = sqrt[90 * 40 * 25 * 25] = sqrt[2250000] = 1500 square meters.

3. Calculate the total cost: Multiply the area by the cost per square meter: Total cost = Area * Cost per square meter = 1500 square meters * Rs 7/square meter = Rs 10500

Correct Answer: Rs 10500

Ans: C

Solution:
Since P, Q, and R are midpoints, RP is parallel to AC and RP = AC/2. Also, RQ is parallel to BC and RQ = AB/2. The perimeter of ARPQ is AR + RP + PQ + QA. Because PQ is parallel to AB and PQ = AB/2 and AR = AB/2 and AQ = AC/2. Perimeter = AB/2 + AC/2 + AB/2 + AC/2. So, The perimeter of ARPQ is AR + RQ + QP + PA. So perimeter= 30/2+ 21/2 + 30/2+21/2= 15+21/2+15+21/2 = 15+15+21/2+21/2=30+21=51.

Answer: 51

Ans: B

Explanation:
1. Find the radii: The area of a circle is given by πr², where r is the radius.
* Smaller circle: 962.5 = πr₁² => r₁² = 962.5/π ≈ 306.49. Therefore, r₁ ≈ √306.49 ≈ 17.5 cm
* Larger circle: 1386 = πr₂² => r₂² = 1386/π ≈ 441. Therefore, r₂ ≈ √441 = 21 cm

2. Calculate the width of the ring: The width of the ring is the difference between the radii of the two circles.
* Width = r₂ – r₁ = 21 cm – 17.5 cm = 3.5 cm

Correct Option: B

Ans: C

Solution:
Let the equal sides be 5x and the base be 3x. The perimeter is 5x + 5x + 3x = 13x = 40. So, x = 40/13.
The sides are (200/13) cm, (200/13) cm, and (120/13) cm.
To find the area, find the height. Half the base is 60/13.
height = sqrt((200/13)^2 – (60/13)^2) = sqrt((40000 – 3600)/169) = sqrt(36400/169) = (20/13)sqrt(91)
Area = (1/2) * (120/13) * (20/13)sqrt(91) = (1200/169)sqrt(91)

Answer: (1200/169)√91

Ans: B

Explanation: First, convert the speed from km/hr to cm/min. Then, calculate the circumference of the wheel. Next, determine how many centimeters the wheel travels per revolution. Finally, divide the speed in cm/min by the distance per revolution to find the revolutions per minute (RPM).

1. Convert Speed to cm/min:
* 66 km/hr * (100000 cm / 1 km) * (1 hr / 60 min) = 110000 cm/min

2. Calculate Wheel Circumference:
* Circumference = 2 * pi * radius = 2 * pi * 35 cm ≈ 219.91 cm

3. Calculate RPM:
* RPM = (Speed in cm/min) / (Circumference in cm/revolution)
* RPM = 110000 cm/min / 219.91 cm/revolution ≈ 500.2 RPM

Correct Answer: 500

Ans: B

Solution:
Side of rhombus = 60/4 = 15 m
Half of diagonal 1 = 18/2 = 9 m
Half of diagonal 2 = √(15² – 9²) = √(225 – 81) = √144 = 12 m
Diagonal 2 = 12 * 2 = 24 m
Area of rhombus = (1/2) * d1 * d2 = (1/2) * 18 * 24 = 216 sq m
Area to be fertilized = 2 * 216 = 432 sq m
Cost = 432 * 3 = Rs 1296
Answer: Rs 1296