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<p>The figure shows an ancient coin which was once used in China. The coin is in the shape of a circle of radius $3$ cm with a square of sides $x$ cm removed from its centre. The area of each face of the coin is $7\pi$ cm$^2$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="250" style="width:300px;height:250px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li><div class="list-item-paragraph">Form an equation in $x$ and show that it reduces to $2\pi - x^2=0$.</div></li><li><div class="list-item-paragraph">Solve the equation $2\pi - x^2=0$.</div></li><li><div class="list-item-paragraph">Find the perimeter of the square.</div></li></ol>

The figure shows an ancient coin which was once used in China. The coin is in the shape of a circle of radius $3$ cm with a square of sides $x$ cm removed from its centre. The area of each face of the coin is $7\pi$ cm$^2$.

Properties of Circle

<p>$AB$, $AC$ and $AD$ are chords of a circle such that&nbsp;$AB = 4$ cm, $AC = 10$ cm and $AD = 11$ cm. Given that&nbsp;$AC$ bisects $\angle BAD$, find the area of the circle.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770728619550-uu6wi9exw8l.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3Mjg2MTk1NTAtdXU2d2k5ZXh3OGwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4MywiZXhwIjoxNzg1NzExMTgzfQ.xoPmBNnk2QTL9ZXDqWFqXmHRQfg5rPGPuoN3hp9oGmU" style="max-width: min(160px, 100%); height: auto; display: block;" alt="Chords in a circle" class="alignright custom-width" /></figure></div>

$AB$, $AC$ and $AD$ are chords of a circle such that $AB = 4$ cm, $AC = 10$ cm and $AD = 11$ cm. Given that $AC$ bisects $\angle BAD$, find the area of the circle.

IP2U14 HOT Q5

<p>Given that the radius of the semicircle is $6$ cm, $ABCD$ and $CPQR$ are squares such that&nbsp;$A$, $Q$ are on the circumference, $B$, $C$, $P$ are on the diameter. Find the total area of the squares $ABCD$ and $CPQR$.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770728173829-4yattd8463o.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3MjgxNzM4MjktNHlhdHRkODQ2M28ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4MywiZXhwIjoxNzg1NzExMTgzfQ._tzRBHqbyns0XUiLpH_pv3VygM-_WWo5hAb2JY2_g7w" style="max-width: min(220px, 100%); height: auto; display: block;" alt="A semicircle containing two squares, ABCD and CPQR, sharing a common side DC and RQ on a line perpendicular to the diameter BP." class="alignright custom-width" /></figure></div>

Given that the radius of the semicircle is $6$ cm, $ABCD$ and $CPQR$ are squares such that $A$, $Q$ are on the circumference, $B$, $C$, $P$ are on the diameter. Find the total area of the squares $ABCD$ and $CPQR$.

IP2U14 HOT Q4

<p>$ABCD$ is a quadrilateral such that&nbsp;$\angle BAD={{56}^{{}^\circ }}$ and $\angle ADC={{108}^{{}^\circ }}$. $DC$ is produced to $P$ such that&nbsp;$\angle BPD={{52}^{{}^\circ }}$ and $BC$ bisects $\angle ABP$. If&nbsp;$\angle BDC={{35}^{{}^\circ }}$, find $\angle ACB$ and $\angle ACP$.</p>

$ABCD$ is a quadrilateral such that $\angle BAD={{56}^{{}^\circ }}$ and $\angle ADC={{108}^{{}^\circ }}$. $DC$ is produced to $P$ such that $\angle BPD={{52}^{{}^\circ }}$ and $BC$ bisects $\angle ABP$. If $\angle BDC={{35}^{{}^\circ }}$, find $\angle ACB$ and $\angle ACP$.

IP2U14 HOT Q3

<p>$P$, $Q$ and $R$ are points that lie on a circle. The tangent to the circle at $R$ meets the line $PQ$ produced at $S$. Point $X$ is on the line $PQ$ such that&nbsp;$RX$ bisects $\angle PRQ$. Prove that&nbsp;$RS=XS$.</p>

$P$, $Q$ and $R$ are points that lie on a circle. The tangent to the circle at $R$ meets the line $PQ$ produced at $S$. Point $X$ is on the line $PQ$ such that $RX$ bisects $\angle PRQ$. Prove that $RS=XS$.

IP2U14 HOT Q2

<p>$A$, $D$, $E$ and $F$ are points on the circle such that&nbsp;$AD = ED$. $ADB$, $AEC$ and $DFC$ are straight lines and $CD$ bisects $\angle BCE$.</p>
<p>$\angle BED = 34^\circ$ and $\angle BFC = 124^\circ$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Prove that</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$AB$ is perpendicular to $CD$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">$BCED$ is a cyclic quadrilateral.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Describe clearly another circle that will pass through $A$, $C$ and $D$.</span></div></li></ol>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770727124566-2340bxfgz36.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3MjcxMjQ1NjYtMjM0MGJ4Zmd6MzYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4MywiZXhwIjoxNzg1NzExMTgzfQ.Kk26B5DFd5w-oLABmY29tTkh5vRkEfn1-b87WPhyVuw" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

$A$, $D$, $E$ and $F$ are points on the circle such that $AD = ED$. $ADB$, $AEC$ and $DFC$ are straight lines and $CD$ bisects $\angle BCE$.

$\angle BED = 34^\circ$ and $\angle BFC = 124^\circ$.

Describe clearly another circle that will pass through $A$, $C$ and $D$.

IP2U14 HOT Q1

<p>Two circles touch each other at $A$. The chord $MN$ of the larger circle cuts the smaller circle at $P$ and $Q$. Prove that&nbsp;$\angle NAQ = \angle MAP$.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770726548885-zot8xkegjz.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3MjY1NDg4ODUtem90OHhrZWdqei5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwODgzLCJleHAiOjE3ODU3MTExODN9.1Gt7nReddhtCmZt2KBrtfZzbKC2jFx_kLSxvxl_W8LI" style="max-width: min(150px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

Two circles touch each other at $A$. The chord $MN$ of the larger circle cuts the smaller circle at $P$ and $Q$. Prove that $\angle NAQ = \angle MAP$.

IP2U14 Exercise Q4

<p>Chords $AC$ and $BD$, perpendicular to each other, intersect at $E$.</p>
<p>In $\Delta ADE$, the altitude from $E$ meets $AD$ at $F$. $FE$ produced meets $BC$ at $G$. Prove that&nbsp;$G$ is the midpoint of $BC$.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770726144466-1u5dr35xwkc.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3MjYxNDQ0NjYtMXU1ZHIzNXh3a2MucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4MywiZXhwIjoxNzg1NzExMTgzfQ.2AcOlE0HS8TreqOiAmS21XV3pD6HYsYUL61wXgpH7P8" style="max-width: min(170px, 100%); height: auto; display: block;" alt="Geometric construction of perpendicular chords and an altitude within a circle" class="alignright custom-width" /></figure></div>

Chords $AC$ and $BD$, perpendicular to each other, intersect at $E$.

In $\Delta ADE$, the altitude from $E$ meets $AD$ at $F$. $FE$ produced meets $BC$ at $G$. Prove that $G$ is the midpoint of $BC$.

IP2U14 Exercise Q3

<p>$A$, $B$, $C$, $D$ and $E$ are points on the circle such that&nbsp;$AC$ is perpendicular to $BE$ and $EC$ is perpendicular to $AD$. Prove that&nbsp;$\angle BCD$ is twice of $\angle ACE$.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770725840886-skhnb16n6x8.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3MjU4NDA4ODYtc2tobmIxNm42eDgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4MywiZXhwIjoxNzg1NzExMTgzfQ.08mNYdM2tVWJhNRGzXX9CTNdkNgNF7Vw2iOfwlce8zQ" style="max-width: min(160px, 100%); height: auto; display: block;" alt="Points A, B, C, D, and E on a circle with intersecting chords AC, BE and EC, AD." class="alignright custom-width" /></figure></div>

$A$, $B$, $C$, $D$ and $E$ are points on the circle such that $AC$ is perpendicular to $BE$ and $EC$ is perpendicular to $AD$. Prove that $\angle BCD$ is twice of $\angle ACE$.

IP2U14 Exercise Q2

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: center;"><p><strong>Circle Symmetrical Property $3$:</strong></p><p>The tangent at the point of contact is <strong>perpendicular</strong> to the radius of the circle. (i.e $TAN \perp OA$)</p></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1770725402720-jyxvv62f3z.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3MjU0MDI3MjAtanl4dnY2MmYzei5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwODgzLCJleHAiOjE3ODU3MTExODN9.8FHZbq93pekz4OCP5-UybhsjtTJlSm0JCYgikc4k-vQ" style="max-width: min(170px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

IP2U14 14.1 Symmetrical Properties of Circle 3 (Proof of Tan. perp. Radius)

<p>$A$, $M$, $B$, $C$ and $N$ are points on a circle. $BN$ and $MC$, the angle bisectors of $\angle ABC$ and $\angle ACB$ respectively, intersect at $P$. Prove that the point $N$ is equidistant from $A$ and $P$.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770724970454-60j7w6kdkqe.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3MjQ5NzA0NTQtNjBqN3c2a2RrcWUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4MywiZXhwIjoxNzg1NzExMTgzfQ.xeJ20AE-2Vw8PkC463pqJvb8NGuuXcyLBjvv76bgm_4" style="max-width: min(150px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

$A$, $M$, $B$, $C$ and $N$ are points on a circle. $BN$ and $MC$, the angle bisectors of $\angle ABC$ and $\angle ACB$ respectively, intersect at $P$. Prove that the point $N$ is equidistant from $A$ and $P$.

IP2U14 Exercise Q1

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>The diagram shows a circle $ABC$, with centre $O$.<br>$AP$ and $BP$ are tangents to the circle.<br>$\angle OAB = 25{}^\circ$</p><ol class="abc-list"><li><div class="list-item-paragraph">Stating your reasons clearly, find</div><ol class="i-list"><li><div class="list-item-paragraph">$\angle AOB$.</div></li><li><div class="list-item-paragraph">$\angle PAB$.</div></li><li><div class="list-item-paragraph">$\angle APB$.</div></li></ol></li><li><div class="list-item-paragraph">Given that the radius of the circle is $5$ cm and $OP=16$ cm, find the length of $AP$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767602723698-5zgahl0huoa.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDI3MjM2OTgtNXpnYWhsMGh1b2EucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4MywiZXhwIjoxNzg1NzExMTgzfQ.CbEKHLtV1e8BZARYE-RfeoKjW7Bg1LaMQaMW8zezDYs" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

Angle Properties of Circle

Solving for Variables in Intersecting Circle Lines and Tangents

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>In the diagram below, $O$ is the centre of the circle and points $A$, $B$, $C$ and $D$ lie on the circumference of the circle.<br>The tangent at $A$ meets $CD$ produced at $E$.<br>The line $AC$ passes through $O$ and lines $AC$ intersects $BD$ at point $F$.<br>It is also given that&nbsp;$\angle ACD=46{}^\circ$ and $\angle CAB=25{}^\circ$.<br>Stating your reasons clearly, find</p><ol class="abc-list"><li><div class="list-item-paragraph">$\angle ADO$.</div></li><li><div class="list-item-paragraph">$\angle OFD$.</div></li><li><div class="list-item-paragraph">$\angle AEC$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767602520657-fzpds21ague.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDI1MjA2NTctZnpwZHMyMWFndWUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4MywiZXhwIjoxNzg1NzExMTgzfQ.iNX7Lod4_wjiPeq3fzq0ozoJ3gjmCPexXhwV08JtjQQ" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The Geometry of Inscribed Triangles and Cyclic Polygons

<p>In the figure, $AE$ is a tangent to the circle whose centre is $O$ and $ABC$ is a straight line.<br>$\angle BAE = 38{}^\circ$, $\angle BOE = 72{}^\circ$ and $\angle DCE = 40{}^\circ$.<br>Find, stating your reasons clearly</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">$\angle ECB$,</div></li><li><div class="list-item-paragraph">$\angle OEC$,</div></li><li><div class="list-item-paragraph">$\angle CED$,</div></li><li><div class="list-item-paragraph">$\angle OBA$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767602304815-a9qazn86x5.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDIzMDQ4MTUtYTlxYXpuODZ4NS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwODgzLCJleHAiOjE3ODU3MTExODN9.KFHpEd0vRMJAqw0lzd6tdGQb1y6jGQuIjeskVIPvuog" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the figure, $AE$ is a tangent to the circle whose centre is $O$ and $ABC$ is a straight line.
$\angle BAE = 38{}^\circ$, $\angle BOE = 72{}^\circ$ and $\angle DCE = 40{}^\circ$.
Find, stating your reasons clearly

Determining Angular Values from Diameters, Chords, and Arcs

<p>In the diagram, $A$, $B$, $C$ and $D$ are points on the circumference of the circle centre $O$.<br>$ABE$ and $DCE$ are straight lines and $ST$ is a tangent to the circle at $A$.<br>$ST$ is parallel to $DE$, $\angle CEB=36{}^\circ$ and $\angle DAB=83{}^\circ$.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">Calculate</div><ol class="i-list"><li><div class="list-item-paragraph">angle $OAB$,</div></li><li><div class="list-item-paragraph">angle $ABC$,</div></li><li><div class="list-item-paragraph">angle $AOD$.</div></li></ol></li><li><div class="list-item-paragraph">Show that&nbsp;$ACD$ is an isosceles triangle.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767601914948-dxhsqkds96b.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDE5MTQ5NDgtZHhoc3FrZHM5NmIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4MywiZXhwIjoxNzg1NzExMTgzfQ.YMFzB5LcMT0WJX7M7B_r6ItIkuNkubRURPgS7PH8xjM" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the diagram, $A$, $B$, $C$ and $D$ are points on the circumference of the circle centre $O$.
$ABE$ and $DCE$ are straight lines and $ST$ is a tangent to the circle at $A$.
$ST$ is parallel to $DE$, $\angle CEB=36{}^\circ$ and $\angle DAB=83{}^\circ$.

Investigating Geometric Proofs through Circle Angle Invariants

<p>In the diagram, $O$ is the centre of the circle through $A$, $B$, $C$ and $D$ and $TB$ is the tangent to the circle at $B$.<br>The diameter $AC$ and the chord $BD$ intersects at $X$.<br>Given&nbsp;$\angle DCX = 32{}^\circ$ and $\angle OBC = 40{}^\circ$, calculate, stating your reasons clearly,</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">$\angle OAB$,</div></li><li><div class="list-item-paragraph">$\angle CXD$,</div></li><li><div class="list-item-paragraph">$\angle ABT$,</div></li><li><div class="list-item-paragraph">$\angle BAD$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767601627209-u1wmorjuuwo.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDE2MjcyMDktdTF3bW9yanV1d28ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4MywiZXhwIjoxNzg1NzExMTgzfQ.4TLTeUO4ci6Iv4mQnqPDP2k0DJXiTVC8snvtC2NisSc" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the diagram, $O$ is the centre of the circle through $A$, $B$, $C$ and $D$ and $TB$ is the tangent to the circle at $B$.
The diameter $AC$ and the chord $BD$ intersects at $X$.
Given $\angle DCX = 32{}^\circ$ and $\angle OBC = 40{}^\circ$, calculate, stating your reasons clearly,

Euclidean Geometry: Interior Angles and Circular Arc Relationships

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>In the diagram, $O$ is the centre of the circle.<br>$PQ$ is parallel to $SR$, $\angle PRS=42{}^\circ$ and $\angle OSR=28{}^\circ$.<br>Calculate</p><ol class="abc-list"><li><div class="list-item-paragraph">$\angle POS$,</div></li><li><div class="list-item-paragraph">$\angle PQR$,</div></li><li><div class="list-item-paragraph">$\angle OPR$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767601424278-uidoomrf8j.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDE0MjQyNzgtdWlkb29tcmY4ai5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwODgzLCJleHAiOjE3ODU3MTExODN9.d7lI1edeSJ0rojLQMc4d3s8MnZUBDD1G-AoQTLfjbWo" style="max-width: min(170px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

Theoretical Foundations of Subtended Angles and Arcs

<p>In the circle with centre $O$, $AD$ is a diameter.</p>
<p>The points $B$ and $C$ lie on the circumference of the circle such that&nbsp;$\angle ADB = 28^\circ$ and $\angle BDC = 42^\circ$.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Given that&nbsp;$BCE$ is a straight line, find</p><ol class="abc-list"><li><div class="list-item-paragraph">$\angle ABD$,</div></li><li><div class="list-item-paragraph">$\angle BAC$,</div></li><li><div class="list-item-paragraph">$\angle DCE$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767601123722-sf58xhxg18h.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDExMjM3MjItc2Y1OHhoeGcxOGgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4MywiZXhwIjoxNzg1NzExMTgzfQ.p4GjAPSd7PY_GhBsuc3gOKGNChzpZUBl3EKusMggze8" style="max-width: min(220px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the circle with centre $O$, $AD$ is a diameter.

The points $B$ and $C$ lie on the circumference of the circle such that $\angle ADB = 28^\circ$ and $\angle BDC = 42^\circ$.

Angles in the Same Segment of a Circle: Equality Proofs

<p>Find the value of all the unknowns in each diagram, stating your reasons clearly.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600941766-ve646enj0k.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDA5NDE3NjYtdmU2NDZlbmoway5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwODgzLCJleHAiOjE3ODU3MTExODN9.EOKrOdp_rfJFLdgtu7lrbBl2jFNcaSmAdF-vxFAceJk" alt="List item image" class="alignleft custom-width" style="max-width: min(170px, 100%); height: auto;" /></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600949603-hw00mfenavm.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDA5NDk2MDMtaHcwMG1mZW5hdm0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDg4MywiZXhwIjoxNzg1NzExMTgzfQ.MDDDbu3wBwp8IlfJCSQl9HmPO7RhuKjrYUfLFftPvxg" alt="List item image" class="alignleft custom-width" style="max-width: min(170px, 100%); height: auto;" /></div></li></ol></div></div></div>

Find the value of all the unknowns in each diagram, stating your reasons clearly.

Geometric Invariants within Circular Segments

<p>Find the value of all the unknowns in each diagram, stating your reasons clearly.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600680009-piavagfuvo.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDA2ODAwMDktcGlhdmFnZnV2by5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwODgzLCJleHAiOjE3ODU3MTExODN9.RLQYrHpKW7Bl9hdQ2Zc8q-oEOOE0_9qq_KlBISC0qPM" alt="List item image" class="alignleft custom-width" style="max-width: min(180px, 100%); height: auto;" /></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600690575-tcaxv2nr0w.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDA2OTA1NzUtdGNheHYybnIwdy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwODgzLCJleHAiOjE3ODU3MTExODN9.-99v2uNDQFye8B7VYLOCg9v7ehW7EdL2nOFxaq0lKbU" alt="List item image" class="alignleft custom-width" style="max-width: min(170px, 100%); height: auto;" /></div></li></ol></div></div></div>

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