Circle or Semi Circle MCQ Quiz - Objective Question with Answer for Circle or Semi Circle - Download Free PDF

Given:

Length of the chord = 12 cm

Perpendicular distance from the center of the circle to the chord = 5 cm

Formula Used:

Radius of the circle can be calculated using the relationship between the chord, perpendicular distance, and radius:

\(r^2 = \left(\frac{\text{Chord Length}}{2}\right)^2 + \text{Perpendicular Distance}^2\)

Calculation:

Chord Length / 2 = 12 / 2 = 6 cm

Perpendicular Distance = 5 cm

Substitute values into the formula:

⇒ r² = 6² + 5²

⇒ r2 = 36 + 25

⇒ r2 = 61

Find r:

⇒ r = √61

⇒ r ≈ 7.81 cm

The radius of the circle is approximately 7.81 cm.

Calculation:

Calculate the Circumference (Perimeter) of the Circular Ground:

Total cost of fencing = ₹440

Cost of fencing per meter = 25 paise = ₹0.25

Circumference = 440 / 0.25 = 1760 meters

Calculate the Radius of the Circular Ground:

The formula for the circumference of a circle is C = 2πr, where r is the radius.

1760 = 2 × (22/7) × r

1760 = (44/7) × r

r = (1760 × 7) / 44

r = 40 × 7

r = 280 meters

Calculate the Area of the Circular Ground:

The formula for the area of a circle is A = πr².

A = (22/7) × (280)²

A = (22/7) × 280 × 280

A = 22 × 40 × 280

A = 880 × 280

A = 246400 square meters

Calculate the Cost of Cutting the Grass:

Cost of cutting grass = 60 paise per 100 square meters.

Cost per square meter = 60 paise / 100 = 0.6 paise = ₹0.006

Total cost of cutting grass = Area × Cost per square meter

Total cost = 246400 × 0.006

Total cost = ₹1478.40

∴ The cost of cutting the grass is ₹1478.40.

Given:

Radius of the circle = 14 cm

Radius of the semi-circle = 14 cm

Formula Used:

Perimeter of a circle = 2 × π × radius

Perimeter of a semi-circle = π × radius + diameter

Calculation:

Perimeter of the circle = 2 × π × 14

Perimeter of the circle = 28π

Perimeter of the semi-circle = π × 14 + 2 × 14

Perimeter of the semi-circle = 14π + 28

Ratio of the perimeters = (Perimeter of the circle) / (Perimeter of the semi-circle)

⇒ Ratio = (28π) / (14π + 28)

⇒ Ratio = 28π / (14(π + 2))

⇒ Ratio = 2π / (π + 2)

Approximating π = 3.14:

Ratio = (2 × 3.14) / (3.14 + 2)

⇒ Ratio = 6.28 / 5.14

⇒ Ratio ≈ 1.22 : 1

Correct Ratio ≈ 12 : 9

The ratio of their perimeters is 12 : 9.

CONCEPT:

Intersection of Circles:

• Two circles intersect when their distance from the centers is less than the sum of their radii but greater than the absolute difference of their radii. In the case of two circles with known equations, their intersection points are the solutions of the system of equations representing the circles.
• The equation of a circle with center (h, k) and radius r is given by:

  (x - h)^2 + (y - k)^2 = r^2

• For two circles to intersect, their equations are solved simultaneously to find common points (x, y) that satisfy both equations.

EXPLANATION:

• We are given two circles:
  • Circle 1: Center at (0.5, 0) with radius 0.5, equation: (x - 0.5)² + y² = 0.25.
  • Circle 2: Center at (1, 1) with radius 1, equation: (x - 1)² + (y - 1)² = 1.

• By solving these two equations simultaneously, we find the intersection points. Solving for x and y, we find one of the points of intersection as (0.2, 0.4).
• The other intersection point is (1, 0), but since we are looking for the first intersection point, the correct answer is (0.2, 0.4).

Therefore, the first intersection point is (0.2, 0.4).

Given:

Original order: One 12-inch circular pizza

Formula used:

Area of a circle = \(\pi r^2\) or \(\pi (\frac{d}{2})^2\) = \(\frac{\pi d^2}{4}\), where r = radius, d = diameter

Calculations:

Original pizza area (12-inch diameter):

⇒ Area = \(\frac{\pi \times 12^2}{4}\) = \(\frac{144\pi}{4}\) = \(36\pi\) square inches

Option 1: Six 4-inch pizzas

⇒ Area of one 4-inch pizza = \(\frac{\pi \times 4^2}{4}\) = \(\frac{16\pi}{4}\) = \(4\pi\) square inches

⇒ Total area = 6 × \(4\pi\) = \(24\pi\) square inches

Option 2: Four 6-inch pizzas

⇒ Area of one 6-inch pizza = \(\frac{\pi \times 6^2}{4}\) = \(\frac{36\pi}{4}\) = \(9\pi\) square inches

⇒ Total area = 4 × \(9\pi\) = \(36\pi\) square inches

Option 3: Seven 3-inch pizzas

⇒ Area of one 3-inch pizza = \(\frac{\pi \times 3^2}{4}\) = \(\frac{9\pi}{4}\) square inches

⇒ Total area = 7 × \(\frac{9\pi}{4}\) = \(\frac{63\pi}{4}\) = \(15.75\pi\) square inches

Option 4: Five 5-inch pizzas

⇒ Area of one 5-inch pizza = \(\frac{\pi \times 5^2}{4}\) = \(\frac{25\pi}{4}\) square inches

⇒ Total area = 5 × \(\frac{25\pi}{4}\) = \(\frac{125\pi}{4}\) = \(31.25\pi\) square inches

∴ The best value for her money is Four 6-inch pizzas.

Given:

Diameter of semicircle = 14√2 cm

Radius = 14√2/2 = 7√2 cm

Total no. of chords = 6

Concept:

Since the chords are equal in length, they will subtend equal angles at the centre. Calculate the area of one sector and subtract the area of the isosceles triangle formed by a chord and radius, then multiply the result by 6 to get the desired result.

Formula used:

Area of sector = (θ/360°) × πr²

Area of triangle = 1/2 × a × b × Sin θ

Calculation:

The angle subtended by each chord = 180°/no. of chord

⇒ 180°/6

⇒ 30°

Area of sector AOB = (30°/360°) × (22/7) × 7√2 × 7√2

⇒ (1/12) × 22 × 7 × 2

⇒ (77/3) cm²

Area of triangle AOB = 1/2 × a × b × Sin θ

⇒ 1/2 × 7√2 × 7√2 × Sin 30°

⇒ 1/2 × 7√2 × 7√2 × 1/2

⇒ 49/2 cm²

∴ Area of shaded region = 6 × (Area of sector AOB - Area of triangle AOB)

⇒ 6 × [(77/3) – (49/2)]

⇒ 6 × [(154 – 147)/6]

⇒ 7 cm²

∴ Area of shaded region is 7 cm²

Given : 

Length of an arc of a circle is 4.5π.

Area of ​​the sector circumscribed by it is 27π cm2.

Formula Used : 

Area of sector = θ/360 × πr²

Length of arc = θ/360 × 2πr

Calculation : 

According to question,

⇒ 4.5π = θ/360 × 2πr 

⇒ 4.5 = θ/360 × 2r   -----------------(1)

⇒ 27π = θ/360 × πr2 

⇒ 27 = θ/360 × r2       ---------------(2)

Doing equation (1) ÷ (2)

⇒ 4.5/27 = 2r/πr²

⇒ 4.5/27 = 2/r

⇒ r = (27 × 2)/4.5

⇒ Diameter = 2r = 24

∴ The correct answer is 24.

Given:

Radius of the wheel of car = 14 cm

Speed of car = 132 km/hr

Formula Used:

Circumference of the wheel = \(2\pi r\) 

1 km = 1000 m

1m = 100 cm

1hr = 60 mins.

Calculation:

Distance covered by the wheel in one minute = \(\frac{132 \times 1000 \times 100}{60}\) = 220000 cm.

Circumference of the wheel = \(2\pi r\) = \(2\times \frac{22}{7} \times 14\) = 88 cm

∴ Distance covered by wheel in one revolution = 88 cm

∴ The number of revolutions in one minute = \(\frac{220000}{88}\) = 2500.

∴ Therefore the correct answer is 2500.

Given:

∠ROS = 42º

Concept used:

The sum of the angles of a triangle = 180°

Exterior angle = Sum of opposite interior angles

Angle made by an arc at the centre = 2 × Angle made by the same arc at any point on the circumference of the circle

Calculation:

Join RQ and RS

According to the concept,

∠RQS = ∠ROS/2

⇒ ∠RQS = 42°/2 = 21°   .....(1)

Here, PQ is a diameter.

So, ∠PRQ = 90°  [∵ Angle in the semicircle = 90°]

In ΔRQT, ∠PRQ is an exterior angle

So, ∠PRQ = ∠RTQ + ∠TQR

⇒ 90° = ∠RTQ + 21°  [∵ ∠TQR = ∠RQS = 21°]

⇒ ∠RTQ = 90° - 21° = 69°

⇒ ∠PTQ = 69°

∴ The measure of  ∠PTQ is 69°

Given:

AB is a diameter of a circle with a center O

∠APC = 62º

Concept used:

The radius/diameter of a circle is always perpendicular to the tangent line.

Sum of all three angles of a triangle = 180°

Calculation:

Minor arc AC will create angle CBA

∠APC = 62º = ∠APB

∠BAP = 90° (diameter perpendicular to tangent)

In Δ APB,

∠APB + ∠BAP + ∠PBA = 180° 

⇒ ∠PBA = 180° - (90° + 62°)

⇒ ∠PBA = 28° 

∴ The measure of minor arc AC is 28° 

Mistake PointsMeasure of the minor arc AC is asked,

∠ABC marks arc AC, 

∴ ∠ABC is the correct angle to show a measure of arc AC

This is a previous year's question, and according to the commission, this is the correct answer.

Given:

Length of an arc = 23.1 cm

Angle subtended on center by arc = 18°

Formula used:

Length of an arc = (2 × π × R × θ)/360

Area of circle = π × R²

Where, R = radius

Calculation:

Length of an arc = (2 × π × R × θ)/360

⇒ 23.1 = (2 × 22 × R × 18)/(360 × 7)

⇒ 23.1 = (22 × R)/(10 × 7)

⇒ R = (2.1 × 70)/2 = 73.5 cm

Area of circle = π × R2

⇒ (22/7) × 73.5 × 73.5

⇒ 22 × 10.5 × 73.5

⇒ 16978.50 cm2

∴ The correct answer is 16978.50 cm2.

The two sides holding the right angle in a right-angled triangle are 3 cm and 4 cm long,

⇒ Length of hypotenuse = (3² + 4²)^1/2 = 5 cm

⇒ Radius of circum-circle = 5/2 = 2.5 cm

Given:

Diameter of pizza = 28cm

Formula:

Circumference of circle = πd

Calculation:

Radius of pizza = 28/2 = 14cm

Total circumference of pizza = 22/7 × 28 = 88cm

Circumference of 3/4 of pizza = 88 × 3/4 = 66cm

∴ Perimeter of remaining pizza = 66 + 14 + 14 = 94cm

Given:

The circumference of the two circles is 198 cm and 352 cm respectively.

Concept used:

Circumference of the two circles = 2πr

Where, r = radius

Calculation:

Let the radius of two circle is r₁ & r₂ 

According to the question,

2πr2 - 2πr1 = 352 - 198

⇒ 2π(r2 - r1) = 154

⇒ π(r2 - r1) = 77

⇒ r2 - r1 = 77 × 7/22

⇒ r2 - r1 = 49/2

⇒ r2 - r1 = 24.5

∴ The required answer is 24.5 cm

Given:

A play ground has a circular path with a certain width around it.

Difference between the circumference of the outer and inner circle is 144 cm

Formula used:

Circumference of a circle = 2πr unit

where r → radius of the circle.

Calculation:

Let the inner radius and outer radius be r cm and R cm respectively.

The width of the path will be (R - r) cm

Difference between the circumference of the outer and inner circle = 144 cm

⇒ 2πR - 2πr = 144

⇒ 2π(R - r) = 144

⇒ R - r = (144 × 7)/44

⇒ R - r = 22.9 ≈ 23

∴ The width of the path is 23 cm.