Question

For each of the statements given below, say true or false:

1. Any two triangles that are congruent to each other will also be similar to each other
2. If in a triangle with sides a, b and c, if a² + b² > c² the triangle has to be acute-angled.
3. Any parallelogram inscribed inside a circle has to be a rectangle.
4. If in a triangle the orthocenter, incenter and circumcenter are collinear the triangle has to be isosceles.
5. There will be a unique circle passing through any three points.

Explanatory Answer

1. TRUE


2. FALSE
   This is true only if c is the largest side. Thinking about it differently, for any triangle, in sum of the squares of the two larger sides will be greater than the square of the smallest side. So, this would imply that any triangle has to be acute angled.


3. TRUE
   Opposite angles of a parallelogram are equal.
   Opposite angles of a cyclic quadrilateral are supplementary.


4. TRUE
   In any triangle, orthocenter, circumcenter and centroid will be collinear. So, if orthocenter, incenter and circumcenter are collinear, then all 4 points lie on the same straight line, which implies the triangle has to be isosceles.


5. FALSE
   There is a unique circle passing through any three non-collinear points.