Methods Of Construction And Symmetries

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Methods Of Construction And Symmetries

page revision: 0, last edited: 15 Sep 2008 08:58

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The unmarked bold line is referred to as "CD" in the text.

ReplyOptionsIn my 2006? printing, figure 108 point $C'$ seems to be labeled $C$ (sans prime).

ReplyOptionsExercise 207:

Construct a triangle, given: (a) its base, the altitude, and a lateral side; (b) its base, the altitude, and an angle at the base; (c) an angle, and two altitudes dropped to the sides of this angle; (d) a side, the sum of the other two sides, and the altitude dropped to tone of these sides; (e) an angle at the base, the altitude, and the perimeter.(a)

Use a compass and straightedge to construct segment, $AB$, congruent to the given base. Extend the segment in the direction of $B$, a length congruent to $AB$, terminal point $A'$. At an arbitrary point $D$ along $AA'$ erect perpendicular $CD$ using the method in $\S 65$. Use the method in $\S 65$ to erect a perpendicular to $CD$ at $C$ and extend it in both directions, line $YZ$. $YZ$ and $AA'$ are both perpendicular to $CD$ and parallel, $\S 71$) two perpendiculars to the same line are parallel.

Set the compass step congruent to the given lateral side, place the pin at $B$, and sweep until it intersects $YZ$ twice, points $F$ and $F'$. Connect $B$ and $F$ by line, and $B$ and $F'$ by line, $BF = BF' = given lateral$.

$\triangle FBF'$ is isoceles, $\S 33$) isoceles triangle has two congruent sides, and $\angle BFF' = \angle BF'F$, $\S 35(2)$ base angles are congruent in an isoceles triangle.

$\angle BF'F = \angle F'BA'$ and $\angle BFF' = \angle FBA$, $\S 77(1)$ parallel lines corresponding angles are congruent.

$\angle F'BA' = \angle FBA$. $\triangle AFB = \triangle A'F'B$, $\S 40(1)$ SAS.

$\triangle AFB$ or $\triangle A'F'B$ is the required triangle.

(b)

Use a compass and straigtedge to construct the base segment, $AB$, congruent to the given base. Use the method in $\S 65$ to erect a perpendicular at the midpoint of $AB$, perpendicular $CD$. Use the method in $\S 65$ to erect a perpendicular to $CD$ at point $C$, perpendicular $YZ$. $YZ \parallel AB$, $\S 71$) two perpendiculars to the same line are parallel.

Use the method in $\S 63$ to construct an angle congruent to the given angle at point $A$ with one side along $AB$ and the other side on the same side of $AB$ as $YZ$; the latter side's intersection with $YZ$ is point $E$.

Construct line $EB$. $\triangle EAB$ is the required triangle.

(c)

Let the given altitudes be $CD$ and $EF$. Construct a line segment $AZ$. Use the method in $\S 63$ to construct an angle congruent to the given angle at point $A$ with one side along $AZ$ and the remaining side is $AM$. Use the method in $\S 65$ to erect a perpendicular congruent to $CD$ at point $A$ and in the same part of the place as $AM$, perpendicular $AC'$. Use the same method to erect a perpendicular to $AM$ at point $M$, congruent to $EF$, and in the same part of the plane as $AZ$, perpendicular $ME'$.

Use the method in $\S 65$ to erect a perpendicular to $AC'$ at $C'$, perpendicular $C'H$, and erect a perpendicular to $E'M$ at $E'$, perpendicular $E'J$. The intersection point of these perpendiculars if $G$. $C'H$ intersects $AM$ at point $K$. $E'J$ intersects $AZ$ at $L$. Construct line $KL$. $\triangle AKL$ is the required triangle.

(d)

Use a compass and straightedge to construct a segment congruent to the given sum of the other two sides, $AB$. Use the method in $\S 65$ to erect a perpendicular to $A$. Mark a segment congruent to the given altitude, originating at $A$, along this perpendicular, segment $AC$. Again use the method in $\S 65$ to erect a perpendicular to $AC$ at $C$ and in the direction of $B$, perpendicular $CZ$.

Set the compass step congruent to the given side, place the pin at $A$, and sweep until it intersects $CZ$, intersection point $D$. Construct line $DB$.

Use the method in $\S 63$ to construct an angle at $D$ congruent to $\angle DBA$ with one leg along $DB$ and the other in the direction of $AB$. The latter intersects $AB$ at point $E$. $DE = BE$, $\S 45(1)$ sides opposite to congruent angles are congruent. $\triangle ADE$ is the required triangle.

(e)

Use a compass and straightedge to construct a segment $AB$ congruent to the given perimeter. Use the method in $\S 65$ to erect a perpendicular to $AB$ at $A$. Mark a segment along this perpendicular congruent to the altitude, originating at $A$; segment $AC$. Use the method in $\S 65$ to erect a perpendicular to $AC$ at $C$ and in the direction of $B$, $CZ$. Use the method in $\S 63$ to construct an angle at $A$ congruent to the given angle with a side along $AB$ and the other in the direction or $CZ$. The latter intersects $CZ$ at point $D$. $AD$ is a side of the triangle.

Subtract side $AD$ from the perimeter segment by marking a segment along $AB$ originating at $B$ and in the direction of $A$, terminal point $E$. Construct line $DE$.

Use the method in $\S 63$ to construct an angle congruent to $\angle DEF$ at point $D$ with one side along $DE$ and the other in the direction of $AB$. The latter intersects $AB$ at point $F$. $DF = EF$, $\S 45(1)$ sides opposite to congruent angles are congruent. $\triangle ADF$ is the required triangle.

ReplyOptionsExercise 208:

Construct a quadrilateral given three of its sides and both diagonals.Use a compass and straightedge to construct a segment of the quadrilateral congruent to the largest given segment, $C'D'$. Set the compass step congruent to given side $AB$, place the pin at $D'$, and mark an arc. Set the compass step congruent to given diagonal $BD$, place the pin at $C'$, and mark an arc. The intersection of these two arcs is point $Z$.

Set the compass step congruent to given side $AD$, place the pin at $C'$, and mark an arc. Set the compass step congruent to given diagonal $AC$, place pin at $D'$, and mark an arc. The intersection of these two arcs is point $Y$.

Construct line segments $C'Y$, $YZ$, and $D'Z$. $C'YZD'$ is a quadrilateral, $C'D' = CD$, $C'Y = AD$, $YZ = BC$, $D'Z = AB$, $C'Z = BD$, $D'Y = AC$.

ReplyOptionsExercise 209:

Construct a parallelogram given: (a) two non-congruent sides and a diagonal; (b) one side and both diagonals; (c) the diagonals and the angle between them; (d) a side, the altitude, and a diagonal (is this always possible?).(a)

Use a compass and a straightedge to construct a side $C'D'$ congruent to given side $CD$. Set the compass step congruent to given diagonal $EF$, place the pin at $C'$, and mark an arc. Set the compass step congruent to given side $AB$, place the pin at $D'$, and mark an arc. The intersection point of the previous two arcs is $B'$. Connect $D'$ and $B'$ by line.

Use the method in $\S 63$ to construct an angle congruent to $\angle C'D'B'$ at point $B'$ with one side along $D'B'$ in the direction opposite $D'$, and the other side in the same part of the plane as $C'D'$. The latter angle side is $B'M$. Mark a segment congruent to $C'D'$ along $B'M$, originating at $B'$ and in the direction of $M$, terminal point $N$. Construct line $C'N$.

$\angle C'D'B' = \angle NB'Z$, $C'D' \parallel NB'$; $\S 73(1)$ two lines with congruent corresponding angles on a transverse are parallel. $C'D' = NB'$ by construction. $C'D'B'N$ is a parallelogram, $\S 86(2)$ quadrilateral with two opposite sides parallel and congruent is a parallelogram.

(b)

Use a compass and a straightedge to construct a side $A'B'$ congruent to the given side. $\S 87(1)$ parallelogram diagonals bisect each other. Use the method in $\S 67$ to bisect the given diagonals $CD$ and $EF$, midpoints $G$ and $H$ respectively. Set the compass step congruent to $CG$, place the pin at $A'$, and mark an arc. Set the compass step congruent to $EH$, place the pin at $B'$, and mark an arc. The intersection point of these two arcs, $Z$, is the intersection point of the diagonals in the parallelogram.

Construct a segment congruent to $CD$, originating at $A'$, and through $Z$; terminal point $D'$. Construct a segment congruent to $EF$, originating at $B'$, and through $Z$; terminal point $F'$.

Construct line segments $D'F'$, $B'D'$, $A'F'$. $A'B'D'F'$ is the required parallelogram.diagonals in the parallelogram.

Construct a segment congruent to $CD$, originating at $A'$, and through $Z$; terminal point $D'$. Construct a segment congruent to $EF$, originating at $B'$, and through $Z$; terminal point $F'$.

Construct line segments $D'F'$, $B'D'$, $A'F'$. $A'B'D'F'$ is the required parallelogram.

(c)

Use a compass and straightedge to construct a segment $C'D'$ congruent to given diagonal $CD$. Use the method in $\S 67$ to bisect this segment, the midpoint is $Z$. Also bisect given diagonal $EF$, its midpoint is $Y$.

Use the method in $\S 63$ to construct an angle congruent to the given angle at point $Z$ with one side along $ZD$. The other side is $ZX$. Extend $ZX$ into the plane on both sides of $CD$. Mark segment $ZV$ congruent to $EY$ along side $ZX$. Mark segment $ZW$ congruent to $EY$ along side $ZW$. $\S 87(1)$ parallelogram diagonals bisect each other.

Construct segments $C'W$, $WD$, $C'V$, $DV$. $C'WDV$ is the required parallelogram.

(d)

Use a compass and straightedge to construct side $A'B'$ congruent to given side $AB$. Use the method in $\S 65$ to erect a perpendicular at $A'$. Mark a segment $A'D'$ congruent to given altitude $CD$ along this perpendicular. Use the method in $\S 65$ to erect a perpendicular at $D'$ into the area of the plane containing $B'$, perpendicular $D'M$.

Set the compass step congruent to given diagonal $EF$, place the pin at $A'$, and sweep until it intersects $D'M$; intersection point is $Z'$. Mark a segment $Z'Y'$ congruent to $AB$ along $Z'D'$. $\S 71$) two lines perpendicular to the same line are parallel. $\S 84 + \S 85$) parallelogram has pairwise parallel and congruent sides. $A'B' = Y'Z'$ by construction, $A'B' \parallel Y'Z'$ by construction.

Construct segments $A'Y'$ and $B'Z'$. $A'Y'Z'B'$ is the required parallelogram, $\S 86(2)$ quadrilateral with two opposite sides parallel and congruent is a parallelogram.

This construction assumed the altitude was from a parallelogram where the given side is one of the bases. If the altitude were to the unknown sides, it seems that it might be impossible to determine the angle between the given side and its adjacent sides.

ReplyOptionsExercise 210:

Construct a rectangle, given a diagonal and the angle between the diagonals.The unknown diagonal is congruent to the given diagonal, $\S 90$. This is the same problem as #209(c).

ReplyOptionsExercise 211:

Construct a rhombus given: (a) its side and a diagonal; (b) both diagonals; (c) the distance between two parallel sides, and a diagonal; (d) an angle, and the diagonal passing through its vertex; (e) a diagonal, and an angle opposite it; (f) a diagonal, and the angle it forms with one of the sides.(a)

All sides are congruent to the given side, $\S 91$) rhombus is a parallelogram with all congruent sides, diagonals bisect the angles and are perpendicular to each other. Use a compass and straightedge to construct a diagonal segment $AB$ congruent to the given diagonal. Use the method in $\S 67$ to bisect $AB$, midpoint $M$. Use the method in $\S 65$ to erect a perpendicular to $AB$ at $M$, perpendicular $CD$. Extend $CD$ into both sides of the plane about $AB$.

Set the compass step congruent to the given side, place the pin at $A$, and sweep an arc. This arc intersects $CD$ at two points, $W$ and $X$. Construct line segments $AW$, $WB$, $BX$, and $AX$. $AWBX$ is the required rhombus.

(b)

Construct a diagonal segment congruent to the first given diagonal, $AB$, using a compass and straightedge. Use the method in $\S 67$ to bisect $AB$, midpoint is $M$. Also bisect the second given diagonal, $CD$, midpoint is $N$. Use the method in $\S 65$ to erect a perpendicular to $AB$ at point $M$, perpendicular $MZ$. Extend this perpendicular into both sides of the plane about $AB$.

Mark a segment $MD$ congruent to $CN$ along $MZ$. Mark a segment $ME$ congruent to $CN$ along $MZ$, originating at $M$ and in the direction away from $Z$. Construct segments $AD$, $DB$, $BE$, $AE$. $ADBE$ is the required rhombus, $\S 91$) rhombus is a parallelogram with perpendicular diagonals that bisect each other.

(c)

Construct a line segment $AB$. Use the method in $\S 67$ to bisect $AB$, midpoint $C$. Use the method in $\S 65$ to erect a perpendicular to $AB$ at point $A$, perpendicular $AC$. Mark a segment $AD$ congruent to the given distance between sides along $AC$. Use the method in $\S 65$ to erect a perpendicular to $AD$ at point $D$, perpendicular $DE$. $AB \parallel DE$ and two sides of the rhombus are along these lines. $\S 51$) distance from point to a line is a perpendicular, $\S 71$) two perpendiculars to the same line are parallel.

Set the compass step congruent to the given diagonal, place the pin at point $A$, and sweep an arc until an intersection with $DE$, point $F$. Construct diagonal segment $AF$.

Use the method in $\S 67$ to bisect $AF$, midpoint $G$. Use the method in $\S 65$ to erect a perpendicular to $AF$ at point $G$, perpendicular $GH$. Extend $GH$ on both sides of $AF$ until it intersects $DE$, intersection point $J$, and intersects $AB$, intersection point $K$. Construct segments $AJ$ and $FK$. $AJFK$ is the required rhombus, $\S 91$) rhombus diagonals are perpendicular and bisect each other.

(d)

Construct line segment $AB$. Use the method in $\S 63$ to construct an angle congruent to the given angle at $A$ with one side along $AB$ and the other side with a point $C$. Use the method in $\S 64$ to bisect $\angle ABC$, bisector $AZ$. Mark a segment $AD$ congruent to the given diagonal along $AZ$.

Use the method in $\S 67$ to bisect $AD$, midpoint $E$. Use the method in $\S 65$ to erect a perpendicular to $AD$ at point $E$, perpendicular $EF$. Extend $EF$ into the plane on both sides of $AD$ until it intersects $AB$, intersection point $G$, and intersects $AC$, intersection point $H$. Construct line segments $DG$ and $DH$. $AGDH$ is the required rhombus; $\S 91(1)$ rhombus diagonals bisect the angles of the rhombus and bisect each other.

(e)

Use the method in $\S 63$ to construct $\angle ABC$ congruent to the given angle. Continue side $AB$ through the vertex to a point $D$. $\angle ABC + \angle CBD = 2d$, $\S 22$) supplementary angles. In the rhombus to be constructed, $BC$ is a transversal to parallels $AB$ and the side opposite to it. $\angle CBD$ is the other angle in this rhombus, $\S 77(3)$ parallels transversed same side interior angles sum is 2d.

Use the method in $\S 64$ to bisect $\angle CBD$, bisector is $BE$. Set the compass step congruent to the given diagonal, place the pin at $B$, and mark a point $F$ along $BE$. $\S 91$) rhombus diagonals bisect its angles. Use the method in $\S 67$ to bisect diagonal $BF$, midpoint $G$. Use the method in $\S 65$ to erect a perpendicular to $BF$ at point $G$ and in the direction of $D$, perpendicular $GH$. Extend this perpendicular into the other side of the plane toward $C$, its intersection with $BC$ is $J$.

Construct sides $JF$ and $HF$. Diagonals $BF$ and $HJ$ are perpendicular by construction. All the angles about $G$ are right, $\S 22$) supplementary angles. $\angle CBF = \angle DBF$; bisector construction. $\triangle DBG = \triangle CBG$; $\S 40(2)$ ASA. $\triangle DBG = \triangle DFG$; $\S 40(1)$ SAS. $\triangle CBG = \triangle CFG$; $\S 40(1)$ SAS.

Therefore all of these triangles are congruent and the sides of the quadrilateral are congruent. $BDFC$ is the required rhombus; $\S 86(1)$ quadrilateral with congruent opposite sides, $\S 91$) parallelogram with all congruent sides is a rhombus.

(f)

Use a compass and a straightedge to construct diagonal $AB$ congruent to the given diagonal. Use the method in $\S 63$ to construct $\angle CAB$ at point $A$ congruent to the given angle with one side along $AB$. Again do this to construct $\angle DAB$ at point $A$ congruent to $\angle CAB$ with one side along $AB$ and the other into the side of the plane that does not contain $C$. Use this method again to construct $\angle MBA$ at point $B$ congruent to $\angle CAB$ with one side along $BA$. Sides $BM$ and $AC$ intersect at point $Y$.

Use the same method to construct $\angle NBA$ at point $B$ congruent to $\angle CAB$ with one side along $BA$ and the other into the part of the plane that does not contain $M$. Sides $BN$ and $AD$ intersect at point $Z$. $\S 85$) opposite angles in a parallelogram are congruent, $\S 91(1)$ rhombus diagonals bisect the angles.

$\angle YAB = \angle ZAB = \angle YBA = \angle ZBA$ by construction. $\triangle YAB = \triangle ZAB$; $\S 40(2)$ ASA and these are isoceles $\S 33$. $YA = YB = ZA = ZB$; $\S 45(2)$ in a triangle sides opposite to congruent angles are congruent. $YAZB$ is the required rhombus; $\S 86(1)$ quadrilateral with opposite sides congruent is a parallelogram, $\S 91$) parallelogram with all congruent sides is a rhombus.

ReplyOptionsExercise 212:

Construct a square, given its diagonal.$\S 92$) square is a rectangle rhombus. $\S 90(1)$ rectangle diagonals are congruent. $\S 91$) rhombus diagonals are perpendicular and bisect each other.

Use a compass and straightedge to construct diagonal segment $AB$ congruent to the given diagonal. Use the method in $\S 67$ to bisect $AB$, midpoint $C$. Use the method in $\S 65$ to erect a perpendicular to $AB$ at $C$ and extend it into the plane on both sides of $AB$. Mark segments $CD$ and $CE$ congruent to $AC$ originating at $C$ and along both directions of this perpendicular.

Construct sides $AD$, $BD$, $BE$, $AE$. $\angle ACD = \angle BCD = \angle BCE = \angle ACE = d$ by construction; $\S 22$) supplementary angles. $\triangle ACD = \triangle BCD = \triangle BCE = \triangle ACE$; $\S 40(1)$ SAS. $ADBE$ is the required rhombus.

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