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Constructions
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15.1 Introduction
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Geometric figures are constructed with straightedge and compass. Since constructions are based on deductive reasoning, measuring instruments such as the ruler and protractor are not permitted. However, a ruler may be used as a straightedge if its markings are disregarded. In constructions, it is advisable to plan ahead by making a sketch of the situation; such a sketch will usually reveal the needed construction steps. Construction lines should be made light to distinguish them from the required figure. The following constructions are detailed in this chapter: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. To construct a line segment congruent to a given line segment To construct an angle congruent to a given angle To bisect a given angle To construct a line perpendicular to a given line through a given point on the line To bisect a given line segment To construct a line perpendicular to a given line through a given external point To construct a triangle given its three sides To construct an angle of measure 60 To construct a triangle given two sides and the included angle To construct a triangle given two angles and the included side To construct a triangle given two angles and a side not included To construct a right triangle given its hypotenuse and a leg To construct a line parallel to a given line through a given external point To construct a tangent to a given circle through a given point on the circle To construct a tangent to a given circle through a given point outside the circle To circumscribe a circle about a triangle To locate the center of a given circle To inscribe a circle in a given triangle To inscribe a square in a given circle To inscribe a regular octagon in a given circle To inscribe a regular hexagon in a given circle To inscribe an equilateral triangle in a given circle To construct a triangle similar to a given triangle on a given line segment as base
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CHAPTER 15 Constructions
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15.2 Duplicating Segments and Angles
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CONSTRUCTION
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To construct a line segment congruent to a given line segment
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Given: Line segment AB (Fig. 15-1) To construct: A line segment congruent to AB Construction: On a working line w, with any point C as a center and a radius equal to AB, construct an arc intersecting w at D. Then CD is the required line segment.
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Fig. 15-1
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CONSTRUCTION 2:
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Fig. 15-2
To construct an angle congruent to a given angle
Given: jA (Fig. 15-2) To construct: An angle congruent to jA Construction: With A as center and a convenient radius, construct an arc (1) intersecting the sides of jA at B and C. With A , a point on a working line w, as center and the same radius, construct arc (2) intersecting w at B . With B as center and a radius equal to BC, construct arc (3) intersecting arc (2) at C . Draw A C . Then jA is the required angle. (^ABC > ^ A B C by SSS; hence jA > jA .)
SOLVED PROBLEMS
15.1 Combining line segments Given line segments with lengths a and b (Fig. 15-3), construct line segments with lengths equal to (a) a 2b; (b) 2(a b); (c) b a.
Fig. 15-3
Solutions
Use construction 1. (a) On a working line w, construct a line segment AB with length a. From B, construct a line segment with length equal to b, to point C; and from C construct a line segment with length b, to point D. Then AD is the required line segment. (b) Similar to (a). AD a b (a b). b a.
(c) Similar to (a). First construct AB with length b, then BC with length a. AC
CHAPTER 15 Constructions
15.2 Combining angles Given ^ABC in Fig. 15-4, construct angles whose measures are equal to (a) 2A; (b) A (c) B A.
Fig. 15-4