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<p>Is it possible to construct a triangle such that $AB=1$<br>cm, $AC=4$<br>cm and $BC=8.3$<br>cm?<br></p>
<p>If yes, construct such a triangle. If no, state your reasons clearly.</p>

Is it possible to construct a triangle such that $AB=1$
cm, $AC=4$
cm and $BC=8.3$
cm?


If yes, construct such a triangle. If no, state your reasons clearly.

Constructions of Triangles

Geometrical Constructions

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Construct a quadrilateral $ABCD$ such that $AB = 6.3$ cm, $AC = 8.2$ cm, $BD = 5.6$ cm, $AB$ is perpendicular to $BC$, and angle $BAD = 60^\circ$.</span></div><ol class="abc-list"><li><div class="list-item-paragraph">Measure and write down the largest possible size of angle $ADC$.</div></li><li><div class="list-item-paragraph">Construct a straight line parallel to $AB$ that passes through $C$ to meet $AD$ produced at $E$. Measure and write down the length of $CE$ and determine the type of quadrilateral $ABCE$.</div></li></ol></li></ol>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Construct a quadrilateral $ABCD$ such that $AB = 4.5$ cm, $BC = 4$ cm, $CD = 4.2$ cm, diagonal $BD = 3.4$ cm, angle $BAD = 40^\circ$ and angle $ADB$ is acute.</span></div><ol class="abc-list"><li><div class="list-item-paragraph">Measure and write down the length of $AC$.</div></li><li><div class="list-item-paragraph">Construct a point $P$ that lies on side $AD$, such that $P$ is equidistant from $A$ and $B$. Measure and write down the length of $PD$.</div></li></ol></li></ol>

Construct a quadrilateral $ABCD$ such that $AB = 6.3$ cm, $AC = 8.2$ cm, $BD = 5.6$ cm, $AB$ is perpendicular to $BC$, and angle $BAD = 60^\circ$.

Construct a quadrilateral $ABCD$ such that $AB = 4.5$ cm, $BC = 4$ cm, $CD = 4.2$ cm, diagonal $BD = 3.4$ cm, angle $BAD = 40^\circ$ and angle $ADB$ is acute.

Constructions of Quadrillaterals

<ol class="i-list"><li><div class="list-item-paragraph">Construct a parallelogram $ABCD$ such that&nbsp;$AB=11$ cm, $BC=8$ cm and angle $ABC=45^{\circ}$.<br>Measure and write down the length of the diagonal $AC$.</div></li><li><div class="list-item-paragraph">Construct a trapezium $ABCD$ such that&nbsp;$AD$ is parallel to $BC$, $AB=3.7$ cm, $BC=8.8$ cm, $CD=4.2$ cm and angle $ABC=60^{\circ}$.</div><div class="list-item-paragraph">Measure and write down the lengths of $AD$ and the diagonal $AC$.</div></li></ol>

<ol class="i-list"><li><div class="list-item-paragraph">Construct triangle $ABC$ such that&nbsp;$AB=9.3$ cm, $AC=6.5$ cm and angle $BAC=120^{\circ}$.<br>Measure and write down the length of $BC$.</div></li><li><div class="list-item-paragraph">Construct triangle $ABC$ such that&nbsp;$AB=3.5$ cm, angle $ABC=60^{\circ}$ and angle $BAC=90^{\circ}$.<br>Measure and write down the length of $BC$.</div></li></ol>

 Constructions of Triangles

<ol class="i-list"><li><div class="list-item-paragraph">Construct line segment $AB$
 of the length $8$
cm and the perpendicular bisector of $AB$</div></li><li><div class="list-item-paragraph">Construct angle $ABC$
 of the size $128^{\circ}$
 and the angle bisector of angle $ABC$
.</div></li></ol>

Constructions of Bisectors

<p>A quadrilateral $ABCD$ has&nbsp;$\angle BAD=115{}^\circ $, $AB=7$ cm, $AD=5$ cm, $BC=8$ cm and $CD=10$ cm.</p>
<ol class="abc-list"><li data-list-item-id="item-1764331388209-kcpfai665"><div class="list-item-paragraph">Complete accurately the quadrilateral $ABCD$ in the space above.<br><br><br><br><br><br><br><br><br><br><br><br></div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1764331399193-zmkna5d7bh.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQzMzEzOTkxOTMtem1rbmE1ZDdiaC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwNTg1LCJleHAiOjE3ODU3MTA4ODV9.RWA-6jOgyfZoJTxww7w8yOhXZDvjrJHpmIo-adrRmao" alt="List item image" class="aligncenter custom-width" style="max-width: min(320px, 100%); height: auto;" /></div><div class="list-item-paragraph"><br><br><br><br><br><br><br><br><br><br></div></li><li data-list-item-id="item-1764331407972-xdu27hk6f"><div class="list-item-paragraph">Measure and write down the $\angle BCD$.</div></li><li data-list-item-id="item-1764331416986-cf38cpp5j"><div class="list-item-paragraph">Construct</div><ol class="i-list"><li data-list-item-id="subitem-1764331425168-hbkw9l3bj"><div class="list-item-paragraph">the perpendicular bisector of $AB$,</div></li><li data-list-item-id="subitem-1764331427725-7jnki4720"><div class="list-item-paragraph">the angle bisector of $\angle BAD$.</div></li></ol></li><li data-list-item-id="item-1764331452926-28whpiiex"><div class="list-item-paragraph">The perpendicular bisector of $AB$ meets the angle bisector of $\angle BAD$ at $E$.  Mark the<br>point $E$ and write down the length of $AE$.<br></div></li></ol>

A quadrilateral $ABCD$ has $\angle BAD=115{}^\circ $, $AB=7$ cm, $AD=5$ cm, $BC=8$ cm and $CD=10$ cm.

Geometry Constructions: Bisectors in a Quadrilateral

<ol class="abc-list"><li><div class="list-item-paragraph">Draw and label $\Delta ABC$ such that&nbsp;$\angle BAC=48{}^\circ $, $AB=7.9$ cm and $AC=4.8$ cm.</div></li><li><div class="list-item-paragraph">Construct the perpendicular bisector of $AB$.</div></li><li><div class="list-item-paragraph">Construct the bisector of $\angle ABC$.</div></li><li><div class="list-item-paragraph">Label the intersection point of the perpendicular bisector of $AB$ and the angle bisector of $\angle ABC$ as&nbsp;$M$. Measure the distance $M$ from $A$.</div></li></ol>

Triangle Construction and Bisectors

<ol class="abc-list"><li><div class="list-item-paragraph">Construct a quadrilateral $ABCD$ such that&nbsp;$AB=9$ cm, $AD=8$ cm, $BC=6$ cm,$\angle BAD=60{}^\circ $ and $\angle ABC=90{}^\circ $.  </div></li><li><div class="list-item-paragraph">Construct</div><ol class="i-list"><li><div class="list-item-paragraph">the perpendicular bisector of $AD$,</div></li><li><div class="list-item-paragraph">the bisector of $\angle BAD$.</div></li></ol></li><li><div class="list-item-paragraph">A point $P$ is equidistant from $AD$ and $AB$, and is also nearer to $A$ than $D$. Mark and label clearly a possible point $P$.</div></li></ol>

Quadrilateral Construction and Locus Conditions

<p>In triangle $ABC$, $AB=15.7$ cm, $BC=9$ cm and $AC=10$ cm.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Using ruler and compasses only, draw the triangle $ABC$,</div></li><li><div class="list-item-paragraph">Measure the largest angle of the triangle,</div></li><li><div class="list-item-paragraph">Construct</div><ol class="i-list"><li><div class="list-item-paragraph">the bisector of the angle $BAC$,</div></li><li><div class="list-item-paragraph">the perpendicular bisector of the line $AC$.</div></li></ol></li><li><div class="list-item-paragraph">Mark out the point $P$ on your diagram such that&nbsp;$P$ is equidistant from the points $A$ and $C$ and equidistant from the lines $AB$ and $AC$. Measure the length $PA$.</div></li></ol>

In triangle $ABC$, $AB=15.7$ cm, $BC=9$ cm and $AC=10$ cm.

Triangle Construction and Locus Problems

<p>In the triangle $ABC$, $AB=10$ cm, $BC=8$ cm and $\angle BAC=50{}^\circ $. </p>
<ol class="abc-list"><li><div class="list-item-paragraph">Complete two possible triangles $ABC$ that are of different sizes.</div></li><li><div class="list-item-paragraph">Construct</div><ol class="i-list"><li><div class="list-item-paragraph">the perpendicular bisector of $AB$,</div></li><li><div class="list-item-paragraph">the angle bisector of $\angle BAC$.</div></li></ol></li><li><div class="list-item-paragraph">Mark, with an $X$, the point that is equidistant from the points $A$ and $B$, and equidistant from the line $AC$ and $AB$.</div></li><li><div class="list-item-paragraph">Hence, measure the length of $BX$.</div></li></ol>

In the triangle $ABC$, $AB=10$ cm, $BC=8$ cm and $\angle BAC=50{}^\circ $. 

Construction: Two Possible Triangles and a Point Defined by Loci

<ol class="abc-list"><li><div class="list-item-paragraph">Construct quadrilateral $ABCD$ where&nbsp;$AB=12$ cm, $BC=6$ cm, $\angle ABC=90{}^\circ $, $CD=10.5$ cm and $AD=11.4$ cm. </div></li><li><div class="list-item-paragraph">Construct</div><ol class="i-list"><li><div class="list-item-paragraph">the perpendicular bisector of $CD$,</div></li><li><div class="list-item-paragraph">the angle bisector of $\angle ABC$.</div></li></ol></li><li><div class="list-item-paragraph">Mark clearly a possible point which is inside the quadrilateral, equidistant to the points $C$ and $D$, and is nearer to lines $AB$ than $BC$. Label this point $K$.</div></li></ol>

Quadrilateral Construction and Loci

<ol class="abc-list"><li><div class="list-item-paragraph">Construct $\Delta ABC$ in the space provided below, where&nbsp;$AB=8$ cm, $AC=7$ cm and<br>$\angle CAB=60{}^\circ $.  Label the vertices of your triangle clearly.<br></div></li><li><div class="list-item-paragraph">Measure and write down the length of $BC$, giving your answer to $1$ decimal place.</div></li><li><div class="list-item-paragraph">Construct</div><ol class="i-list"><li><div class="list-item-paragraph">the angle bisector of $\angle BCA$,</div></li><li><div class="list-item-paragraph">the perpendicular bisector of $BC$.</div></li></ol></li><li><div class="list-item-paragraph">Point $P$ is a point inside the triangle such that&nbsp;$BP &lt; CP$ and $P$ is located nearer to $BC$ than to $AC$.  Shade the region where&nbsp;$P$ could possibly be located.</div></li></ol>

Triangle Construction and Loci

<p>Construct a quadrilateral $ABCD$ such that&nbsp;$AB=4.5$ cm, $BC=7$ cm, $CD=6$ cm, $AD=5$ cm and the diagonal $AC=8$ cm. Measure and write down the size of angle $ADC$.</p>

Construct a quadrilateral $ABCD$ such that $AB=4.5$ cm, $BC=7$ cm, $CD=6$ cm, $AD=5$ cm and the diagonal $AC=8$ cm. Measure and write down the size of angle $ADC$.

Quadrilateral Construction with Given Side Lengths

<p>Construct a quadrilateral $ABCD$ such that&nbsp;$AB=7.5$ cm, $BC=8$ cm, $CD=6.8$ cm, $AD=4.7$ cm and the diagonal $AC=11$ cm. Measure and write down the size of angle $ACD$.</p>

Construct a quadrilateral $ABCD$ such that $AB=7.5$ cm, $BC=8$ cm, $CD=6.8$ cm, $AD=4.7$ cm and the diagonal $AC=11$ cm. Measure and write down the size of angle $ACD$.

<p>Construct the parallelogram $ABCD$ given the $2$ adjacent sides $AB$ and $BC$ below.</p>
<p>Hence construct the perpendicular bisector of the line $BD$.<br><br><br><br><br><br><br></p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1764329137578-ed9hwpmhwt.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQzMjkxMzc1NzgtZWQ5aHdwbWh3dC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwNTg1LCJleHAiOjE3ODU3MTA4ODV9.E-hCvPQLX_O1cpJb9ow6z3Tfu4FBAco6tFqjSIPJyh4" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

Construct the parallelogram $ABCD$ given the $2$ adjacent sides $AB$ and $BC$ below.

Hence construct the perpendicular bisector of the line $BD$.








Parallelogram Construction and Perpendicular Bisector

<ol class="abc-list"><li><div class="list-item-paragraph">Construct, and label a parallelogram $ABCD$ in which $AB=10$ cm, $AD=4$ cm and angle $BAD=75{}^\circ $.</div></li><li><div class="list-item-paragraph">Construct the angle bisector of angle $ADC$.</div></li><li><div class="list-item-paragraph">Construct the perpendicular bisector of $DC$.</div></li><li><div class="list-item-paragraph">Mark the point at which the angle bisector of angle $ADC$ intersect the perpendicular 	bisector of $DC$ as&nbsp;$X$ and hence measure the reflex angle $AXB$.</div></li></ol>

Parallelogram Construction and Bisectors

<p>Construct a quadrilateral $ABCD$ given&nbsp;$AB=6$ cm, $A\hat{B}C=65{}^\circ $, $BD=8$ cm, $B\hat{A}D=110{}^\circ $ and $A\hat{D}C=95{}^\circ $. On the same diagram, construct</p>
<ol class="abc-list"><li><div class="list-item-paragraph">the bisector of angle $BCD$,</div></li><li><div class="list-item-paragraph">the perpendicular bisector of $BC$.</div></li></ol>

Construct a quadrilateral $ABCD$ given $AB=6$ cm, $A\hat{B}C=65{}^\circ $, $BD=8$ cm, $B\hat{A}D=110{}^\circ $ and $A\hat{D}C=95{}^\circ $. On the same diagram, construct

Construction of Quadrilateral

<ol class="abc-list"><li><div class="list-item-paragraph">Construct a parallelogram $ABCD$ such that, $AB=8$ cm, $BC=6.3$ cm and angle$ABC=62{}^\circ $.</div></li><li><div class="list-item-paragraph">Measure and write down the length of diagonal $AC$. </div></li></ol>

Construction of a Parallelogram

<p>Construct triangle $XYZ$ such that&nbsp;$XY=5$ cm, angle $XYZ=120{}^\circ $ and angle $YXZ=37{}^\circ $. </p>
<ol class="abc-list"><li><div class="list-item-paragraph">Construct the perpendicular bisector of $YZ$.</div></li><li><div class="list-item-paragraph">Construct the angle bisector of angle $YXZ$.</div></li><li><div class="list-item-paragraph">Describe the point of intersection between these two bisectors.</div></li></ol>

Construct triangle $XYZ$ such that $XY=5$ cm, angle $XYZ=120{}^\circ $ and angle $YXZ=37{}^\circ $. 

Triangle Construction and Bisector Constructions

<p>Construct triangle $ABC$ where&nbsp;$BC=6$ cm and $\angle ABC=85{}^\circ $. $AB$ has already been drawn.<br><br><br><br><br><br><br><br><br><br></p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1764327866780-g8v8ijnv1bb.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQzMjc4NjY3ODAtZzh2OGlqbnYxYmIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDU4NSwiZXhwIjoxNzg1NzEwODg1fQ.za4IGot2IPHub5hppttLZcFqa6n5pq1UukcJNyxFejw" style="max-width: min(320px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p><br><br><br><br><br><br><br><br><br><br><br><br><br></p>
<ol class="abc-list"><li data-list-item-id="item-1764327875547-tzp1ssahq"><div class="list-item-paragraph">Construct</div><ol class="i-list"><li data-list-item-id="subitem-1764327898512-mdl49fk2r"><div class="list-item-paragraph">the perpendicular bisector of $AB$.</div></li><li data-list-item-id="subitem-1764327909245-o0ywc53fd"><div class="list-item-paragraph">the bisector of angle $BAC$.</div></li></ol></li><li data-list-item-id="item-1764327877152-urcs1kd37"><div class="list-item-paragraph">Mark clearly a possible point which is inside the triangle, equidistant from $A$ and $B$, and 	is nearer to $AC$ than $AB$. Label this point $P$.</div></li></ol>

Construct triangle $ABC$ where $BC=6$ cm and $\angle ABC=85{}^\circ $. $AB$ has already been drawn.











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