### Geometry Concept testers - Solutions

Given below is the answer key to the set of True of False questions

1. Any two triangles that are congruent to each other will also be similar to each other - TRUE

2. If in a triangle with sides a, b and c, if a^2 + b^2 > c^2 the triangle has to be acute-angled - 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. Any parallelogram inscribed inside a circle has to be a rectangle - TRUE

Opposite angles of a parallelogram are equal

Opposite angles of a cyclic quadrilateral are supplementary

4. If in a triangle the orthocenter, incenter and circumcenter are collinear the triangle has to be isosceles - 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. There will be a unique circle passing through any three points - FALSE

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

6. If two circles with centers A and B and radii r and R intersect, then AB > R - r - TRUE

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

7. Circumradius of a triangle cannot be greater than the three sides of the triangle - 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

8. For any obtuse-angled triangle, the orthocenter and circumcenter will lie outside the triangle - TRUE

9. In a scalene triangle, the sum of the three medians will be greater than the sum of the three altitudes - 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

10. In circumcircle and incircle are concentric, the triangle has to be equilateral - TRUE

Labels: CAT geometry, CAT Solutions