Write the truth value (T/F) of each of the following statements. If possible, give reason for your answer.(i) In an equilateral triangle, the incircle and the circumcircle can be constructed with t...

Question

Write the truth value (T/F) of each of the following statements. If possible, give reason for your answer.
(i) In an equilateral triangle, the incircle and the circumcircle can be constructed with the same centre.

(ii) The incircle of a triangle touches all sides of the triangle.

(iii) In an obtuse-angled triangle, the circumcenter lies on one of the sides of the triangle.

(iv) In an acute-angled triangle, the circumcentre lies inside the triangle.

(v) The incircle of a triangle is constructed by locating the point of intersection of perpendicular bisectors of any two sides of the triangle.

Philoid · Accepted Answer

(i) True. In an equilateral triangle, circumcenter and incenter coincide and hence both the circles can be drawn using the same center.

(ii) True, because while constructing an incircle, a perpendicular is dropped on a side and length of the perpendicular is taken as the radius.

(iii) False. Only in a right-angled triangle does the circumcenter lie on the hypotenuse.

(iv) True. The perpendicular bisectors of the sides intersect inside the triangle.

(v) False. Incenter is the intersection of the angle bisectors of two of the angles.