Understanding Cyclic Quadrilaterals in Geometry
Explore the concept of cyclic quadrilaterals in geometry, understanding their theorems and properties. Learn about circles, naming parts, arcs, sectors, and the relationship between opposite angles in a cyclic quadrilateral. Enhance your problem-solving skills with practical examples and visuals.
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07 May 2025 Cyclic quadrilaterals LO: To solve problems using cyclic quadrilaterals theorems. www.mathssupport.org
Naming the parts of a circle A circle is a set of points equidistant from a fixed point called the centre. The distance around the entire circle boundary is called the circumference. The radius is any line segment joining its centre to any point on the circumference. radius centre The diameter is a line segment passing through the centre. Note that the diameter of a circle is twice its radius circumference www.mathssupport.org
Naming the parts of a circle A chord is any line segment that joins two points on the circle. Therefore, a diameter is an example of a chord. It is the longest possible chord. The line that touches the circumference in exactly one point is called a tangent or a tangent line. The point where the tangent touches the circle is called the point of contact or point of tangency. centre www.mathssupport.org
Naming the parts of a circle The region of a circle enclosed by a chord and an arc a Segment is called Any chord encloses two segments, which have different areas. If the segment is enclosed by the diameter a semicircle Major segment it is called Minor segment www.mathssupport.org
Arcs and sectors arc An arc is any continuous part of the circumference. sector When an arc is bounded by two radii a sector is formed. www.mathssupport.org
Cyclic quadrilateral Four points are said to be concyclic if a circle can be drawn through them If four concyclic points are joined B Draw the line from A to B. Draw the line from B to C. O Draw the line from C to D. C Draw the line from D to A. A A convex quadrilateral is formed. D This quadrilateral is called a Cyclic quadrilateral www.mathssupport.org
Opposite angles in a cyclic quadrilateral Draw a circle and label the centre O. Choose any four points on the circumference and label them A, B, C, D: B Draw the line from A to B. Draw the line from B to C. x Draw the line from C to D. Draw the line from D to A. O Measure the angle ABC (x). Measure the opposite angle ADC (y). w z C A y Add together the angles x and y D What do they add up to? 180o 180o What do the angles w and z add up to? Statement The opposite angles of a cyclic quadrilateral add up to 180o www.mathssupport.org
Opposite angles in a cyclic quadrilateral We have just seen a demonstration that shows that the opposite angles in a cyclic quadrilateral add up to 180 . We can prove this result as follows: Mark the centre of the circle O and label angles ABC and ADC x and y. B x The angles at the centre are 2x and 2y. 2y O (the angle at the centre of a circle is twice the angle at the circumference) 2x C A 2x + 2y = 360 y 2(x + y) = 360 D x + y = 180 www.mathssupport.org
Angles in a cyclic quadrilateral As a result of this theorem we can conclude that if the opposite angles of a quadrilateral add up to 180 , a circle can be drawn through each of its vertices. For example, the opposite angles in this quadrilateral add up to 180 . 67 It is a cyclic quadrilateral. 109 Remember that when two angles add up to 180 they are often called supplementary angles. 113 71 www.mathssupport.org
Angles in a cyclic quadrilateral Find the value of x and y Find the value of x 67 (x + 36) + (x)= 180 (opposite angles in the cyclic quadrilateral) 2x + 36 = 180 2x = 144 (x + 36) x = 72 x y Find the value of y 67 + y = 180 y = 113 www.mathssupport.org
Calculating the size of unknown angles Calculate the size of the labelled angles in the following diagram: a = 64 (angle at the centre) b = c = (180 128 ) 2 = 26 (angles at the base of an isosceles triangle) c d = 33 (angles at the base of an isosceles triangle) f b 128 O d e = 180 2 33 = 114 (angles in a triangle) f = 180 (e + c) = 180 140 = 40 (opposite angles in a cyclic quadrilateral) e a 33 www.mathssupport.org
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