Question

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

1. If two circles with centers A and B and radii r and R intersect, then AB > R - r.
2. Circumradius of a triangle cannot be greater than the three sides of the triangle.
3. For any obtuse-angled triangle, the orthocenter and circumcenter will lie outside the triangle.
4. In a scalene triangle, the sum of the three medians will be greater than the sum of the three altitudes.
5. In circumcircle and incircle are concentric, the triangle has to be equilateral.

Explanatory Answer

1. TRUE
   If two circles intersect, then AB, R and r form sides of a triangle. => r + AB > R => AB > R - r.


2. FALSE
   Think about this differently. Draw a large circle, draw a small triangle inside this circle in a small segment of this circle. The circle is the circumcircle for this triangle.


3. TRUE


4. TRUE
   Shortest distance from a point to a line is the perpendicular distance. => Altitude is always less than or equal to median. In a scalene triangle, altitude will be less than the median.


5. TRUE