Geometry

A chord is a straight line segment whose endpoints both lie on a circle. The perpendicular distance from the center to a chord is a very common exam theme. A line drawn from the center of the circle that hits the chord at a 90-degree angle will always cut that chord into two equal parts. This often creates a right-angled triangle involving the radius, the distance from the center, and half the chord length.

A cyclic quadrilateral is a four-sided figure where all four vertices (corners) lie on the circumference of a circle. The most critical property is that its opposite angles are supplementary, meaning they add up to 180 degrees. If you extend one side of the quadrilateral, the exterior angle formed is equal to the interior opposite angle. This is a favorite topic for UPSC examiners.

Two triangles are similar if they have the same shape but different sizes. This happens when their corresponding angles are equal. When triangles are similar, the ratio of their corresponding sides is constant. Also, the ratio of their areas is the square of the ratio of their sides. For example, if side lengths are in ratio 2:3, the areas will be in ratio 4:9.