Geometry Study Guide

Circles are fundamental shapes in geometry, and it's essential to know the specific names given to their different parts. The centre is the point equidistant from all points on the circle's edge. The radius is a line segment connecting the centre to any point on the circle's circumference. The diameter is a straight line segment passing through the centre and connecting two points on the circumference; it's always twice the length of the radius. The circumference is the total distance around the edge of the circle. A chord is any straight line segment connecting two points on the circumference, which does not necessarily pass through the centre. If a chord passes through the centre, it's a diameter. An arc is a portion of the circumference, like a curved segment. A tangent is a straight line that touches the circle at exactly one point, known as the point of tangency. A sector is a region of the circle bounded by two radii and an arc. A segment is a region bounded by a chord and an arc. These terms are crucial for accurately describing and solving problems involving circles.

The relationship between angles subtended at the centre and at the circumference of a circle is a fundamental concept in circle theorems. When two points on the circumference of a circle (let's call them A and B) define an arc, that arc can 'subtend' angles. An angle subtended at the centre is formed by lines connecting these two points to the centre of the circle (e.g., angle AOB, where O is the centre). An angle subtended at the circumference is formed by lines connecting these two points to any other point on the circumference (e.g., angle ACB, where C is another point on the circumference). The key theorem states that the angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circumference. This relationship holds true regardless of where the point C is located on the major arc, as long as it's on the circumference. This theorem is crucial for calculating unknown angles in circles and forms the basis for understanding other circle theorems, like angles in a semicircle and cyclic quadrilaterals. For example, if angle AOB is 80 degrees, then angle ACB will be 40 degrees. Conversely, if angle ACB is 30 degrees, then angle AOB will be 60 degrees. It's important to identify the arc that subtends both the angle at the centre and the angle at the circumference correctly, paying attention to whether it's the major or minor arc.