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Angles A and C are also supplementary.
Theorem 2. The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle (see figure c). An exterior angle is obtained by extending one of the sides, say BA, to the point E. The theorem states that
Angle SAD = angle BCD
This theorem applies at each vertex of the cyclic quadrilateral when the side is produced to form an exterior angle.
Theorem 3. This theorem is the converse of Theorem 1. It states as follows:
If one pair of opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.
CYCLIC QUADRILATERAL 141
This means that if, in figure d, angle DAB + angle BCD = 180°, then ABCD is a cyclic quadrilateral. It follows that a circle can be drawn through the points A, B, C, and D. For an example of this converse theorem, refer to the entry Angle in a Semicircle.
Theorem 4. Another converse of a theorem can be used to prove that a quadrilateral is cyclic, and is stated as follows (see figure e):
If angle DAC = angle CBD, then the quadrilateral ABCD is cyclic.
This statement is the converse of the theorem stating that the angles on the same arc are equal.
Example. In figure f, O is the center of the circle and angle CAO the sizes of angles ADC and ABC.
Angle ACO = 35° Angle AOC = 110° Angle ADC = 55° Angle ABC = 125°
Triangle AOC is isosceles, because AO and OC are radii Sum of angles of triangle = 180°
Angle at the center is twice the angle at the circumference Sum of opposite angles of a cyclic quadrilateral = 180°
References: Angle at the Center and Circumference of a Circle, Angle Sum of a Triangle, Geometry Theorems, Isosceles Triangle, Quadrilateral, Supplementary Angles.
This is the locus traced by a point on the circumference of a circle when the circle rolls, without slipping, along a line. In the figure the many positions of the point on the rolling circle are joined with a dashed line to show what the cycloid looks like. Imagine the point as being on the tread of a tire as the tire rolls along a flat road.
References: Circle, Locus.
A cylinder is a solid formed by three surfaces. The cylinder described here is made up of two circles in parallel planes joined to a tube formed by rolling up a rectangle (see figure a). The cylinder we shall study is an upright one, so that its axis of symmetry is at right angles to its base; this is usually called a right cylinder. A cylinder is a prism of circular cross section.
The net of the cylinder is made up of a rectangle that rolls up to make a tube, and two circles. Each circle is joined to the rectangle at a point. Thus
Length of the rectangle = circumference of the circle Length = 2ttR
Axis of symmetry
The formulas for the volume and surface area of the cylinder are stated here. The height of the cylinder is H units and the radius of each circular end is R units:
Volume of cylinder = ttR2H cubic units Surface area of cylinder = 2ttRH + 2ttR2 square units
Example. Jack has bought a can of beans from the shop. He measures the dimensions of the can and records them in a sketch as shown in figure b. Find the volume of the can and the area of tin used in its manufacture. Draw a net of the can showing the main dimensions.
D ~ 7.4 cm
H = 11 cm
R = 3.7 cm Volume = ttR2H
= tt x 3.72 x 11 = 473 (to 3 sf)
Radius = one-half of the diameter Formula for the volume of a cylinder. Substituting R = 3.7, H = 11 Using it in the calculator
The volume of the can is 473 cubic centimeters. Next, write
Surface area = 2nRH + 2txR2
= 2 x jt x 3.7 x 11 + 2 x tt x 3.72 = 342 (to 3 sf)
Formula for the surface area of a cylinder
Substituting R = 3.7, H = 11
Using it in the calculator
The surface area of the can is 342 square centimeters.
The net of the can is shown in figure c.
The dimensions L, H, and D of the net that are shown in figure c are calculated in the following way:
C = 2ttR Formula for the circumference of a circle
C = 2 x ^ x 3.7 Substituting R =3.7, which is half of the diameter
C = 23.2 (to 3 sf) Using 7t in the calculator
The length of the rectangle is equal to the circumference of the circular ends: L is 23.2 cm.
The width of the rectangle is equal to the height of the can: H = 11 cm.
The diameter of the circular ends is given as 7.4 cm.
References: Area, Circle, Circumference, Diameter, Prism, Rectangle, Right Angle, Volume.
The set of quantities or information gathered from a sample of a population when a survey or an experiment is done is called the data. Statisticians are people who collect, display, and analyze data in order to draw conclusions and make predictions.
Example. The heights of the students at Washington College is a set of data. The nationalities of the teachers at Washington College is another set of data. Data can be classified according to the data tree shown in the figure: