Book Name |
: Ganit Prakash |

Subject |
: Mathematics |

Class |
: 10 (Madhyamik) |

Publisher |
: Prof Nabanita Chatterjee |

Chapter Name |
: Construction Of Circumcircle And Incircle Of A Triangle (11th Chapter) |

## LET US WORK OUT – 11.1

**Question 1**

**Let us draw the following traingles. By drawing the circumcircle in each case, let us write the position of circumcentre and the length of circumradius by measuring it.**

**(i) An equilateral triangle having each side of length 6 cm.**

**Solution:**

ABC is an equilateral triangle whose each side is 6 cm. Circumcentre O is inside the triangle.

**(ii) An isoceles triangle whose length of the base is 5.2 cm and each of the equal sides is 7 cm.**

**Solution:**

O is the circumcentre of the circumcircle of the PQR, whose length of circum radius is 4 cm. Circumcentre O is inside the triangle.

**(iii) A right-angled triangle having two sides 4 cm and 8 cm length, containing right angle.**

**Solution:**

We draw the circumcircle of the right angled triangle XYZ, whose circumcentre is at the midpoint of the hypotenuse length of circumradius is 4.6 cm

**(iv) ****A right-angled triangle having length of hypotenuse 12 cm and other side of 5 cm length.**

**Solution:**

ABC is a right angled triangle whose length of hypotenuse is 12 cm & length of other side is 5 cm.

Circumcentre O is on the mid point of the hypotenuse.

**(v) A triangle whose dength of one side is 6.7 cm and the two angles adjacent to this side are 75° and 55°.**

**Solution:**

**(vi) ABC is a triangle whose BC = 5 cm, ∠ABC = 100° and AB = 4 cm.**

**Solution:**

Draw the circum circle of ABC, whose centre is outside the triangle & length of circumradius is 3.7 cm.

**Question 2**

**Given PQ = 7.5 cm. ∠QPR = 45°, ∠PQR = 75° ; PQ = 7.5 cm. ∠QPS = 60°, ∠PQS = 60°. Let us draw ∆PQR and ∆PQS in such a way that the points R and S lie on the same side of P Q , let us draw the circumcircle of ∆PQR and let us observe and write the position of the point S within, on and outside the circumcircle. Let us find out its explanation.**

**Solution:**

We draw the circumcircle of the ∆PQR. The point S is on that circumcircle.

**Question 3**

**Given AB = 5 cm, ∠BAC = 30°, ∠ABC = 60°; AB = 5 cm, ∠BAD = 45°, ∠ABD = 45°. Let us draw ∆ABC and ∆ABD in such a way that the points C and D lie on opposite sides of AB. Let us draw the circle circumscribing ∆ABC. Let us write the position of the point D with respect to the circumcircle. Let us write by understanding what there characteristics we are observing here.**

**Solution:**

We draw the circumcircle of ∆ABC. The point D is on the circumcircle. The circumcentre is on the mid point of AB.

**Question 4**

**We draw a quadrilateral ABCD having AB = 4 cm, BC = 7 cm CD = 4 cm,∠ABC = 60°,∠BCD = 60°. Let us draw the circle circumscribing ABC and write what other characteristics we observe.**

**Solution:**

We draw the circumcircle of ∆ABC whose centre O is inside the ∆ABC & inside the quadrilateral ABCD.

**Question 5**

**Let us draw the rectangle PQRS having PQ = 4 cm, and QR = 6 cm. Let us draw the diagonals of the rectangle. Let us write by calculating the position of the centre of the circumcircle of ∆PQR and the length of circumradius without drawing. By drawing the circumcircle of ∆PQR, let us verify.**

**Solution:**

The diagonals PQ & RS of the rectangle PQRS are drawn. The circumcentre of the ∆PQ f is at the intersecting point of the diagonals. The length of the circumradius is 3.8 cm.

**Question 6**

**If any circular picture is given, then how shall we find its centre? Let us find the centre of the circle in the adjoining figure.**

**Solution:**

By drawing, circumcentre & incentre of an equilateral triangle are same.

## LET US WORK OUT – 11.2

**Question 1**

**Let us draw the following triangles and by drawing the incircle of each circle, let us write by measuring the length of inradius in each case.**

**(i) The lengths of three sides are 7 cm, 6 cm and 5.5 cm.**

**Solution:**

Radius of the incircle is 1.8 cm.

**(ii) The length of two sides are 7.6 cm, 6 cm and the angle included by those two sides is 75°.**

**Solution:**

Radius of the incircle is 2 cm.

**(iii) The length of one side is 6.2 cm and measures of two angles adjacent to this side are 50° and 75°.**

**Solution:**

Radius of the incircle is 1.8 cm.

**(iv) A triangle is a right-angled triangle whose two sides containing the right angle have the lengths of 7 cm and 9 cm.**

**Solution:**

Radius of the incircle = 2.2 cm.

**(v) A triangle is an isosceles triangle whose two sides containing the right ang have the lengths of 7 cm and 9 cm.**

**Solution:**

Radius of the incircle =1.8 cm.

**(vi) The triangle is an isoceles triangle having the 7.8 cm. Length of base and the length of each equal side are 6.5 cm and 10 cm.**

**Solution:**

Length of the inradius = 2.1 cm.

**(vii) A triangle is an isosceles triangle having the 10 cm length of base and each of the equal angle is 45°.**

**Solution:**

Length of the radius of the incircle = 2.1 cm.

**(viii) Let us draw an equilateral triangle having 7 cm length of each side. By drawing the circumcircle and incircle, let us find the length of circumradius and inradius and let us write whether there is any relation between them.**

**Solution:**

ABC in an equilateral traingle. Here length of in circle two triangle is 2 cm & length of radius of circumcircle is 4 cm .

\therefore Circumcircle = 2 × in radius