A figure representing the given parts - COLLEGE

Step III: A statement of what is given in the representation of Step II. I Step IV: A statement of what is to be constructed, that is, the ultimate result to be'1

obtained. I.

Step V: The construction, with a descriPtion of each step. An authG in the construction must be given.

Step VI: A proof that the construction in Step V gives the desired results.

Most constructions will involve the intersection properties of two lines, of a:~

line and a circle, or of two circles. In the developments of our constructions,~

we will assume the following: ,j

1. A straight line can be constructed through any two given points (PostUl~!

ate 2).:J

2. It is possible to construct a circle in a plane with a given point Pas cente and a given segmentAB as radius [see Fig. 10.2 (Postulate 19)].

--

GEOMETRIC CONSTRUCTIONS

305

A B

Fig. 10.2.

3. Two coplanar nonparallel lines intersect in a point (Theorem 3.1).

4. Two circles 0 and P with radii a and b intersect in exactly two points if the distance c between their centers is less than the sum of their radii but greater than the difference of their radii. The intersection points will lie in different half-planes formed by the line of centers (Fig. 10.3).

5. A line and a circle intersect in exactly two points if the line contains a point inside the circle.

a b

~ ~

Fig. 10.3. a + b > c. a - b < c.

The student will find the solution to a construction problem easier to follow if, in the solution, he is able to distinguish three different kinds oflines. We will employ the following distinguishing lines:

(a) Given lines, drawn as heavy full black lines.

(b) Construction lines, drawn as light (but distinct) lines.

COnstruction 1

10.4. At a point on a line construct an angle congruent to a given angle.

L c

7f ,// / T

\

S//

iT-j--'R "N

Construction 1.

Given: LABC, line MN, and a point P on

MN.

To construct: An angle congruent to LABC having P as vertex and

PFI

^as

side.

Construction:

ST ATEMENTS REASONS

1. With B as center and any radius, construct an arc intersecting

BA

atE andBC atD.

2. With P as center and radius = BD, construct RT intersecting MN at R.

3. With R as center and a radius = DE, construct an arc intersecting RT at S.

4. Construct ?S.

5. LRPS == LABG.

Proof

ST ATEMENTS

1. Draw ED and RS.

2. BE = PR; BD = PS.

3. ED=RS.

4. L.RPS

₌₌

L.EBD.

5. LRPS

₌₌

_LABC.

Construction 2

1. Postulate 19.

2. Postulate 19.

3. Postulate 19.

4. Postulate 2.

REASONS

1. Postulate 2.

2. § 7.3.

3. § 7.3.

4. S.S.S.

5. § 4.28.

10.5. To construct the bisector of an angle.

Given: LABC.

To construct: The bisector of LABG.

Construction:

STATEMENTS

307

B D A

Construction 2.

REASONS

1. With B as center and any radius,

construct an arc intersecting

lfA

at D and

BC

^atE.

2. With D and E as centers and any radius greater than one-half the distance from D to E, construct arcs intersecting at F.

3. Construct BF.

4. BF is the bisector of LABC.

1. Postulate 19.

2. Postulate 19.

3. Postulate 2.

Proof:

STA TEMENTS

1. Draw DF and EF.

2. BD=BE;DF=EF.

3. BF=BF.

4. L.DBF

₌₌

L.EBF.

5. La == Lf3.

6. :. BFbisects LABC.

Construction 3

REASONS

I. Postulate 2.

2. § 7.3.

3. Theorem 4.1.

4. S.S.S.

5. § 4.28.

6. § 1.19.

10.6. To construct a perpendicular to a line passing through a point on the line.

308

FUNDAMENTALS OF COLLEGE GEOMETRY

Given: Line 1and a point P of the line.

To construct: A line containing P and perpendicular to I.

Construction: (The construction and proof are left to the student. The student will recognize this to be a special case of Construc- tion 2.)

t

I

~

/ I

I I II

1 I I I I t

t lB

Construction 3.

Exercises

1. Draw an obtuse angle. Then, with a given ray as one side,

an angle congruent to the obtuse angle. :

2. Draw two acute angles. Then construct a third angle whose measure is equal to the sum of the measures of the two given angles.

3. Draw a scalene triangle. Then construct three adjacent angles whose measures equal respectively the measures of the angles of the given triangle. Do the three adjacent angles form a straight angle?

~

A B

Ex. 3.

1 4. Draw a quadrilateral. Then construct an angle whose measure is equal~

to the sum of the measures of the four angles of the given quadrilateral':;i 5. Draw two angles. Then construct an angle whose measure is the differ"!

ence of the measures of the given angles. Label the new angle La. ..~

6. Draw an obtuse aI1Kle. Then construct the bisector of the given angle~

Label the bisector

RS.1

7. Construct an angle whose measure is (a) 45, (b) 135, (c) 67t.

8. Draw an acute triangle. Construct the bisectors of the three angles the acute triangle. What appears to be true of the three angle bisecto

GEOMETRIC CONSTRUCTIONS

309

9. Repeat Ex. 8 with a given right triangle.

10. Repeat Ex. 8 with a given obtuse triangle.

11. Draw a vertical line. At a point on this line construct a perpendicular to the line.

12. Using a protractor, draw LABC whose measure is 45. At any point P on side

BA

^construct a perpendicular to /[11. Label R the point where this perpendicular intersects side Be. AtR construct a line perpendicular to

Be. Label S the point where the second perpendicular intersects

iiB. What kind of triangle is t:,SPR?

Construction 4

B 10.7. To construct the perpendicular

bisector of a given line segment.

A Given: Line segmentAB.

To comtruct: The perpendicular bisector of AB.

t I

Construction: Construction 4.

STATEMENTS REASONS

1. With A and B as centers and with a radius greater than one-half AB, construct arcs intersecting at C andD.

2. Construct CD intersecting

AB

at M.

CD

^is ^the perpendicular bisector ofAB.

1. Postulate 19.

2. Postulate 2.

Proof:

'310

STATEMENTS REASONS

1. Draw AC, AD, BC, BD.

2. AC=BC;AD=BD.

3. CD = CD.

4. LACD

LBCD.

5. LADM

LBDM.

6. DiU = DiVl.

7. LADA1

LBDM.

8. AA1

= BM.

9. LAMD

LBMD.

10. ...CD is 1- bisectorAB.

1. Postulate 2.

2. § 7.3.

3. Theorem 4.1.

4. 5.5.5.

5. § 4.28.

6. Theorem 4.1.

7. S.A.S.

8. § 4.28.

9. § 4.28.

10. Definition of 1- bisector.

Construction 5

t

I

11)

I I I

~~A

/

~ I

I

^~//

I

~ I

/.t.~ '

10.8. To construct a perpendicular to a given line from a point not on the line.

Given: Line l and point P not on l.

To construct: A line 1- to l from P.

Construction: CUIi.\tlllction5.

STATEMENTS REASONS

1. Postulate 19.

I. With P as center and any conven- ient radius, construct an arc intersecting-l at A and B.

2. With A and B as centers and a radius whose measure is greater than the measure of one-half segment AB, construct arcs intersecting at C.

3. Construct PC.

4. PC is

1- to line l.

-- Proof (The proof is left to the student.) Hint: See proof of Construction

~

Discussion: Point C can be either on the same side of l as is P or on the opposi

side.

2. Postulate 19.

3. Postulate 2.

311 Exercises

1. Draw a line segment.

congruent segments.

:2. Draw an acute scalene triangle.

angle.

3. Repeat Ex. 2 for an obtuse scalene triangle.

4. Draw an acute scalene triangle. Construct the perpendicular bisectors of the three sides of the triangle.

S. Repeat Ex. 4 for an obtuse scalene triangle.

6. Draw an acute scalene triangle. Construct the three medians of the triangle.

7. Repeat Ex. 6 for an obtuse scalene triangle.

8. Construct a square.

9. Construct an equilateral triangle ABC. From C, construct the angle bisector, altitude, and median. Are these separate segments? If not, which are the same?

By construction, divide the segment into four Construct the three altitudes of the tri-

Construction 6

~ 10.9. Through a given point to con-

struct a line parallel to a given line.

(;iz'en: Line l and point P not on the line.

To construct: A line through Pill.

Construction:

Construction 6. T

STATEMENTS

REASONS

I. Through P construct any line ST intersecting l at R. Label

Sf

so that P is between Sand T.

2. With P as vertex and

PS

as a side

Construct Lf3

₌₌

La.

3. iii III.

1. Postulate 2.

2. § lOA.

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