Chapter – 11 : Construction Of Circumcircle And Incircle Of A Triangle

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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.

IMG 4346

(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.

IMG 4347

(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

IMG 4348

(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.

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(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:

IMG 4354IMG 4369

(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.

IMG 4355

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:

IMG 4356

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:

IMG 4357

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:

IMG 4358

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:

IMG 4359

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:

IMG 4360

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:

IMG 4361

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:

IMG 4362

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:

IMG 4363

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:

IMG 4364

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:

IMG 4365

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:

IMG 4366

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:

IMG 4367

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:

IMG 4368

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

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