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A quadrilateral is a four-sided plane shape. There are many types, but only the trapezoid, parallelogram, rectangle, and square are described here. 1. A TRAPEZOID is a quadrilateral having only two sides parallel. If the other two sides are equal, it is an isosceles trapezoid. BF is the altitude of the trapezoid.
2. A PARALLELOGRAM is a quadrilateral having opposite sides parallel.
a. AB is parallel to CD. b. AC is parallel to BD. c. AD and CB are diagonals. d. Diagonals bisect each other so CO = OB and AO = OD. e. Opposite angles are equal ACD = DBA and CAB = BDC. f. If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram. g. A parallelogram may be constructed if two adjoining sides and one angle are known. 3. A RECTANGLE is a parallelogram having one right angle.
a. ABCD is a parallelogram having one right angle. This, of course, makes all angles right angles. b. AC and BD are diagonals. c. 0 is the midpoint of AC and BD and OB=OC=OD=CIA. d. 0 is equidistant from BC and AD and is also equidistant from AB and CD. e. A rectangle may be constructed if two adjoining sides are known. 4. A SQUARE is a rectangle having its adjoining sides equal.
a. ABCD is a square. b. AC and BD are diagonals. c. 0 is the geometric center of the square. AO =OC=OB=OD. d. 0 is equidistant from all sides. e. A square may be constructed if one side is known. POLYGONS A polygon is a many-sided plane shape. It is said to be regular if all sides are equal and irregular when they are not. Only regular polygons are described here.
Regular Polygons Triangles and quadrilaterals fit the description of a polygon and have been covered previously. Three other types of regular polygons are shown in the illustration. Each one is inscribed in a circle. This means that all vertices of the polygon lie on the circumference of the circle. Note that the sides of each of the inscribed polygons are actually equal chords of the circumscribed circle. Since equal chords subtend equal arcs, by dividing the circumference into an equal number of arcs, a regular polygon may be inscribed in a circle. Also note that the central angles are equal because they intercept equal arcs. This gives a basic rule for the construction of regular polygons inscribed in a circle as follows: To inscribe a regular polygon in a circle, create equal chords of the circle by dividing the circumference into equal arcs or by dividing the circle into equal central angles. Dividing a circle into a given number of parts has been discussed, so construction should be no problem. Since there are 360 degrees around the center of the circle, you should have no problem in determining the number of degrees to make each equal central angle. Problem: What is the central angle used to inscribe a pentagon in a circle? Solution:
Methods for Constructing Polygons The three methods for constructing polygons described here are the pentagon, the hexagon, and the octagon. The PENTAGON is a method to develop the length of a side and departs from the rule given. Radius PB has been bisected to locate point 0. Radius OC has been used to swing an arc CE from the center 0. E is the intersection of arc CE with diameter AB. Chord CE is the length of the side and is transferred to the circle by are EF using chord CE as radius and C as center. The HEXAGON has been developed by dividing the circumference into 6 equal parts. The OCTAGON method has been developed by creating central angles of 90 to divide a circle into 4 parts and bisecting each are to divide the circumference into 8 equal parts,
Circumscribing a Regular Polygon about a Circle Problem: Circumscribe a hexagon about a given circle.
Solution: Step 1. Divide the circumference into a given number of parts. Step 2. At each division point draw a tangent to the circle. The intersection of the tangents forms vertices of the circumscribed polygon. ELLIPSES
AE is the major axis. BD is the minor axis. |
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