**NCERT Class 10 Maths Lab Manual – Areas of Sectors formed at the Vertices of a Triangle**

**Objective**

To verify that sum of areas of three sectors of the same radii ‘r’ formed at the vertices of any triangle is \(\frac {1}{2}\) πr^{2} by paper cutting and pasting.

**Prerequisite Knowledge**

- Concept of different types of triangles.
- Definition of a sector.
- Area of circle = πr
^{2}, r → radius.

**Materials Required**

Glazed paper, sketch pens, fevicol, a pair of scissors, pencil, geometry box.

**Procedure**

- Draw three different types of triangles on a glazed paper as shown in fig. (i).

(i) Equilateral ∆ABC

(ii) Isosceles ∆PQR

(iii) Scalene ∆XYZ

- Cut an equilateral ∆ABC as shown in fig.(ii)

- Taking vertices A, B and C as centres of ∆ABC, draw three sectors of same radii r.

- Cut these three sectors and marked them as 1,2,3 and fill different colours.

- Draw a straight line and mark any point ‘O’ on it. Place three sectors 1,2, 3 adjacent to each other so that the vertices A, B, C coincide with ‘O’ without leaving any gap as shown in fig. (v).

- The same process (steps 1-5) can be taken up with isosceles triangle and scalene triangle fig. (i).

**Observation**

The shape formed on the straight line is a semi circle

∴ area of circle = πr^{2}

∴ area of semicircle = \(\frac {1}{2}\) πr^{2}

**Result**

It is verified that the sum of areas of three sectors of same radii ‘r’ formed at the vertices of any triangle is \(\frac {1}{2}\) πr^{2}.

**Learning Outcome**

The students are able to understand the concept of this activity through paper cutting. It is clear to them that a semicircle is always obtained, whatever be the type of a triangle.

**Activity Time**

- Find the sum of areas of four sectors of same radii ‘r’ formed at the vertices (as centre) of any quadrilateral through activity.

**Hint:**

- Find the sum of areas of five sectors of same radii ‘r’ formed at the vertices (as centre) of any pentagon through activity.

**Hint:**

You can also download **Maths Class 10 NCERT Solutions** to help you to revise complete syllabus and score more marks in your examinations.

**Viva Voce**

**Question 1.**

What is the angle subtended by a circle at centre ?

**Answer:**

360°

**Question 2.**

What is the sum of angles of four sectors of same radii ‘r’ formed at the vertices (as centre) at any quadrilateral ?

**Answer:**

360°

**Question 3.**

If the perimeter of semicircle is 12 cm, find its radius.

**Answer:**

\(\frac {7}{3}\) cm

**Question 4.**

What is the area of a semicircle of radius 2 cm ?

**Answer:**

2π

**Question 5.**

Define a segment of a circle.

**Answer:**

A chord divides a circle in two parts each of which is called a segment of a circle.

**Question 6.**

What is the difference between a sector and a segment of a circle ?

**Answer:**

A sector of a circle is formed by an arc and two radii of the circle, while a segment is one of its part in which a chord divides a circle.

**Question 7.**

Define chords.

**Answer:**

A line segment joining two points on the circumference of a circle is known as a chord.

**Question 8.**

**“A tangent intersects the circle in more than one point”**. Is the statement true or false ?

**Answer:**

False

**Multiple Choice Questions**

**Question 1.**

If the area of a semicircle is 121 cm^{2}, find its radius.

(a) √77 cm

(b) √76 cm

(c) √74 cm

(d) None of these

**Question 2.**

If the perimeter and the area of a circle are numerically equal, then the radius of the circle is

(a) π units

(b) 2 units

(c) 4 units

(d) 5 units

**Question 3.**

Area of a sector of angle P (in degree) of a circle witl radius R is

(a) \(\frac{P}{{180}^{0}}\times 2\pi{R}\)

(b) \(\frac{P}{{180}^{0}}\times 2\pi{R}^{2}\)

(c) \(\frac{P}{{360}^{0}}\times 2\pi{R}\)

(d) \(\frac{P}{{720}^{0}}\times 2\pi{{R}^{2}}\)

**Question 4.**

Area of the sector of a circle with radius 4 cm and angle 30° is

(a) 4.91 cm

(b) 14.9 cm

(c) 4.19 cm

(d) 94.1cm

**Question 5.**

Find the area of a quadrant of circle whose circumference is 22 cm.

(a) \(\frac{77}{8}\) cm^{2}

(b) \(\frac{76}{8}\) cm^{2}

(c) \(\frac{77}{2}\) cm^{2}

(d) \(\frac{77}{4}\) cm^{2}

**Question 6.**

Find the perimeter of the quadrant of a circle of radius 4.2 cm.

(d) 15 cm

(b) 8.4 cm

(c) 12.6 cm

(d) None of these

**Question 7.**

If the radius of a circle is increased by 100% by what percent is the area of the circle increased ?

(a) 200%

(b) 300%

(c) 400%

(d) None of these

**Question 8.**

The perimeter of a sector with radius r and angle θ of a circle is given by

(a) \(d+\frac{2\pi r\theta}{{180}^{0}}\)

(b) \(r+\frac{\pi r\theta}{{180}^{0}}\)

(c) \(\frac{\theta}{{360}^{0}}\times 2\pi r\)

(d) \(2r+\frac{\pi r\theta}{{180}^{0}}\)

**Question 9.**

Area of the sector of a circle with radius 7 cm and angle 120° is

(a) 51.33 cm^{2}

(b) 53.11 cm^{2}

(c) 53.13 cm^{2}

(d) Noneofthese

**Question 10.**

The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

(a) \(\frac{145}{3}\) cm^{2}

(b) \(\frac{514}{3}\) cm^{2}

(c) \(\frac{451}{3}\) cm^{2}

(d) \(\frac{154}{3}\) cm^{2}

Answers

1. (a)

2. (b)

3. (d)

4. (c)

5. (a)

6. (a)

7. (b)

8. (d)

9. (a)

10. (d)

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