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 Fun Trivia: G : Geometry

### Special Sub-Topic: Circle Theorems

 One of the easiest circle theorems to remember is to do with angles in a semi-circle. If two straight lines are drawn from either end of the diameter of a circle and meet at a point on the circumference, what will the angle always be?

90 degrees. Two straight lines from either end of the diameter will always meet at a right angle to each other on the circumference. This applies regardless of the length of the diameter and circumference of the circle.

 When an angle is formed from two points on the circumference and two straight lines drawn from these two points meet at another point on the same circumference, it is said that the angle is ________ by an arc.
Subtended. If an angle is subtended by an arc then the point on the circumference where the straight lines from the two points meet will have a certain angle. This can be measured with a protractor. If a second example of this happens whereby from the same two points on the circumference another meeting point is formed, then that second angle will be equal to the first angle. Angles are equal when subtended by the same arc (a=b etc). It is also possible to say that angles are equal when subtended by the same chord. The difference between a chord and an arc is, whether a straight line is drawn between the two points marked on the circumference.

 What is the geometric name given to a straight line which touches the circumference of a circle at one point only?
Tangent. A tangent is the name of a straight line which touches the circumference of a circle at only one point. A tangent can meet the circumference at any point around a circle.

 When the straight line described above meets the radius or diameter of a circle, what is the size of the angle always formed?
90 degrees. This situation also leads to the two lines, the tangent and the radius, meeting at right angles to one another. Another way to describe this would be that the tangent is perpendicular to the radius of the circle.

 Two points are marked on the circumference of a circle. A straight line is drawn from each of these points which both meet in the exact centre of the circle. Another straight line is drawn from each point and meet at a certain point on the circumference. How much bigger is the angle formed at the centre compared to the angle formed at the circumference?
Double. Angles at the circumference are half the size of the angle formed at the circle's centre when the angles are created from the same two points on the circumference. If the angle at the centre is (x) and the angle at the circumference is (y) then the equation for this rule would be - x=2y.

 When a four-sided shape, where each corner touches the circumference is found inside a circle, it is called a cyclic quadrilateral. What is the rule about the size of the angles that are opposite to each other within this cyclic quadrilateral?
They total 180 degrees. Opposite angles within a cyclic quadrilateral will always total 180 degrees. In total, angles in a quadrilateral tally to make 360 degrees. That is the proof of the rule. As there are always two sets of opposing angles in a quadrilateral, each must total 180 degrees and subsequently reach the full 360 degrees.

 A point is marked outside the confines of the circle and from that two straight lines are drawn so that they touch the opposite sides of the circles. What does this tell us about the straight lines from the point of origin to the point of meeting the circumference?
They are equal in length. When this phenomenon occurs, the length of the two straight lines from the point of origin to the points on the circumference of the circle that they touch will be equidistant.

 An important factor to take into consideration when attempting to find angles using circle theorems is the use of isosceles triangles. Which of the following statements is true of isosceles triangles?
Both of these. The use of isosceles triangles when attempting to ascertain angles within a circle makes the process a whole lot easier. When one of the triangle's corners is situated directly on the circle's geometric centre, it is possible to know that two sides (the radii of the circle) will be equidistant. An isosceles triangle is indicated by a dash drawn over each of the two equal sides. Once the isosceles triangle is known, the missing angle can be established. As the angle between the point at where the two equidistant lines meet is always the different sized angle, you know that the other two angles will be equal. If angle x = 50 degrees, angle y = 50 degrees. That will leave a final angle of 80 degrees, as angles in a triangle always amount to 180 degrees.

 One significant rule regarding the attainment of angle sizes within a circle is the 'alternate _______ theorem'.
Segment. The alternate segment theory proposes that two angles in a certain structure will be equal to each other. Take angles (x) and (y). Angle (x) is situated between a chord (a line connecting two points of an arc) and a tangent. Angle (y) is situated between the meeting points of two straight lines which have been drawn from either end of the chord. This meeting point is on the circumference. When angles (x) and (y) are found in this setup they are equal in size. The equation would be x=y.

 We can't possibly do a math topic and not put in an equation, so here we have it! What is the equation for finding the area of a sector within a circle? (r = radius, A = angle, x = multiply, / = divide, pi = 3.14)
pi x r x r x (A/360). pi x r x r x (A/360), is much easier to say than it is to write down (pi times r squared times A divided by 360). Pi is the number which indicates the ratio between the length of the diameter and the length of the circumference. The reason that the angle (A) is divided by the number 360 is because there are 360 degrees in a full circle.

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