Question

Draw any equilateral triangle. Draw incircle and circumcircle of it. What did you observe while doing this activity? 

i. While drawing incircle and circumcircle, do the angle bisectors and perpendicular bisectors coincide with each other? 

ii. Do the incentre and circumcenter coincide with each other? If so, what can be the reason of it? 

iii. Measure the radii of incircle and circumcircle and write their ratio.

Answer 1

Steps of construction: 

i. Construct equilateral ∆XYZ of any measurement. 

ii. Draw the perpendicular bisectors of side XY and side YZ of the triangle. 

iii. Draw the bisectors of ∠X and ∠Z. 

iv. Name the point of intersection of the perpendicular bisectors and angle bisectors as point I. 

v. With I as centre and IM as radïus, draw a circle which touches all the three sides of the triangle. 

vi. With I as centre and IZ as radius, draw a circle which passes through the three vertices of the triangle. 

[Note: Here, point of intersection of perpendicular bisector and angle bisector is same.]

i. Yes. 

ii. Yes. 

The angle bisectors of the angles and the perpendicular bisectors of the sides of an equilateral triangle are coincedent. Hence, its incentre and circumcentre coincide. 

iii. Radius of circumcircle = 3.6 cm, 

Radius of incircle = 1.8 cm 

Ratio = Radius of circumcircle/Radius of incircle = 3.6/1.8 = 2/1 = 2 : 1.