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PSLE

PSLE Foundation Mathematics

<p>\(ABCD\) is a convex 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 convex 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 \(\angle ACP\).

Angles in the Same Segment

Proving Concyclicity

IP2U14 HOT Q3

Properties of Circle

<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 data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Prove that</div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">\(AB\) is perpendicular to \(CD\),</div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">\(BCED\) is a cyclic quadrilateral.</div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Describe clearly another circle that will pass through \(A\), \(C\) and \(D\).</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=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3MjcxMjQ1NjYtMjM0MGJ4Zmd6MzYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTYxMiwiZXhwIjoxNzkyNzI1OTEyfQ.htY-bV0aPy-lpCu5xRzPdot-nrq9WXXw-ZY7NBde_30" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>
<p>As shown in the diagram, \(B,E,F\) are collinear, with \(F\) between \(B\) and \(E\).</p>

\(A\), \(D\), \(E\) 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 \(D\).

As shown in the diagram, \(B,E,F\) are collinear, with \(F\) between \(B\) and \(E\).

Angles in Cyclic Quadrilaterals

Properties of Special Triangles

Angles in a Semicircle and the Converse

IP2U14 HOT Q1

Polygons

<p>In the diagram, the two circles touch each other externally at \(F\) and \(CFT\) is the common tangent. Given that&nbsp;\(ABC\), \(CDE\) and \(AFE\) are straight lines, prove that</p>
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<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">angle \(CBF + \text{angle}\hspace{0.5em} CDF = 180^\circ\), and</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">angle \(BCD + \text{angle}\hspace{0.5em} BFD = 180^\circ\).</div></li><li data-list-item-id="item-1770117831518-ug1rdrv6i" data-list-item-path="item-1770117831518-ug1rdrv6i"><div class="list-item-paragraph">Hence prove that there exists a circle that passes through \(B\), \(C\), \(D\) and \(F\).</div></li></ol>
<p>Based on the answers above, or otherwise, prove that</p>
<ol class="i-list" start="4"><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">triangles \(BCD\) and \(ECA\) are similar, and</div></li><li data-list-item-id="item-v" data-list-item-path="item-v"><div class="list-item-paragraph">there exists a circle that passes through \(A\), \(B\), \(D\) and \(E\).</div></li></ol>

In the diagram, the two circles touch each other externally at \(F\) and \(CFT\) is the common tangent. Given that \(ABC\), \(CDE\) and \(AFE\) are straight lines, prove that

angle \(CBF + \text{angle } CDF = 180^\circ\), and

angle \(BCD + \text{angle } BFD = 180^\circ\).

Based on the answers above, or otherwise, prove that

triangles \(BCD\) and \(ECA\) are similar, and

there exists a circle that passes through \(A\), \(B\), \(D\) and \(E\).

Alternate Segment Theorem

Similarity

Proofs in Plane Geometry

<p>In the diagram, \(TA\) and \(TC\) are tangents from \(T\) to the circle at \(A\) and \(C\), respectively. \(B\) is a point on the circle and the tangent to the circle at \(B\) intersects \(TC\) and \(TA\) at \(D\) and \(E\), respectively. Prove that</p>
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<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">angle \(EDT = 2 \times \text{angle}\hspace{0.5em} BAC\) ,</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">angle \(ABC = 90^\circ + \frac{1}{2} \times \text{angle}\hspace{0.5em} DTE\) , and</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">quadrilateral \(ACDB\) is NOT a cyclic quadrilateral, i.e., there exists no such a circle that passes through \(A\), \(C\), \(D\) and \(B\).</div></li></ol>

In the diagram, \(TA\) and \(TC\) are tangents from \(T\) to the circle at \(A\) and \(C\), respectively. \(B\) is a point on the circle and the tangent to the circle at \(B\) intersects \(TC\) and \(TA\) at \(D\) and \(E\), respectively. Prove that

angle \(EDT = 2 \times \text{angle } BAC\) ,

angle \(ABC = 90^\circ + \frac{1}{2} \times \text{angle } DTE\) , and

quadrilateral \(ACDB\) is NOT a cyclic quadrilateral, i.e., there exists no such a circle that passes through \(A\), \(C\), \(D\) and \(B\).

Triangle Angle Sum and Exterior Angles

<p>In the diagram, \(TA\) and \(TB\) are tangents from \(T\) to the circle at \(A\) and \(B\), respectively. \(C\) is a point on the circle and \(D\) lies on \(BC\) such that&nbsp;\(CD = AD\). Prove that</p>
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<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">triangles \(TAB\) and \(DAC\) are similar,</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">angle \(ATB + \text{angle}\hspace{0.5em} ADB = 180^\circ\), and</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">quadrilateral \(ADBT\) is a cyclic quadrilateral, i.e., there exists a circle that passes through \(A\), \(D\), \(B\) and \(T\).</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">Hence, or otherwise, prove that&nbsp;\(TD\) is parallel to \(AC\).</div></li></ol>

In the diagram, \(TA\) and \(TB\) are tangents from \(T\) to the circle at \(A\) and \(B\), respectively. \(C\) is a point on the circle and \(D\) lies on \(BC\) such that \(CD = AD\). Prove that

triangles \(TAB\) and \(DAC\) are similar,

angle \(ATB + \text{angle } ADB = 180^\circ\), and

quadrilateral \(ADBT\) is a cyclic quadrilateral, i.e., there exists a circle that passes through \(A\), \(D\), \(B\) and \(T\).

Hence, or otherwise, prove that \(TD\) is parallel to \(AC\).

Angles Formed by Two Parallel Lines and a Transversal

Equal Tangents from an External Point

<p>In the diagram, \(TA\) and \(TB\) are tangents from \(T\) to the circle with centre \(O\) at \(A\) and \(B\), respectively. \(AC\) is a diameter of the circle. Prove that</p>
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<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">angle \(AOB = 2 \times\) angle \(ACB\),</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">\(CB\) is parallel to \(OT\),</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">triangles \(OBC\) and \(TAB\) are similar, and</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">quadrilateral \(AOBT\) is a cyclic quadrilateral, i.e., there exists a circle that passes through \(A\), \(O\), \(B\) and \(T\).</div></li></ol>

In the diagram, \(TA\) and \(TB\) are tangents from \(T\) to the circle with centre \(O\) at \(A\) and \(B\), respectively. \(AC\) is a diameter of the circle. Prove that

triangles \(OBC\) and \(TAB\) are similar, and

quadrilateral \(AOBT\) is a cyclic quadrilateral, i.e., there exists a circle that passes through \(A\), \(O\), \(B\) and \(T\).

Angles at the Centre and Circumference

Tangents Perpendicular to Radii

<p>In the diagram, \(SAT\) is the tangent to the circle at \(A\). \(ADB\) and \(AEC\) are straight lines such that&nbsp;\(B\) and \(C\) are on the circle and \(DE\) is parallel to \(SAT\). Prove that</p>
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<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">triangles \(ABC\) and \(AED\) are similar, and</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">quadrilateral \(BDEC\) is a cyclic quadrilateral, i.e., there exists a circle that passes through \(B\), \(D\), \(E\) and \(C\).</div></li></ol>

In the diagram, \(SAT\) is the tangent to the circle at \(A\). \(ADB\) and \(AEC\) are straight lines such that \(B\) and \(C\) are on the circle and \(DE\) is parallel to \(SAT\). Prove that

triangles \(ABC\) and \(AED\) are similar, and

quadrilateral \(BDEC\) is a cyclic quadrilateral, i.e., there exists a circle that passes through \(B\), \(D\), \(E\) and \(C\).

<p>In the figure, \(TA\) and \(TB\) are tangents to the circle, centre \(O\), from \(T\).</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">Given that&nbsp;\(AOC\) is a diameter of the circle, state a reason why \(\angle AOB=2\angle ACB\).</div></li></ol><p>Hence, prove that</p><ol class="i-list" start="2"><li><div class="list-item-paragraph">\(CB\) is parallel to \(OT\),</div></li><li><div class="list-item-paragraph">A circle with diameter \(OT\) can be drawn to pass through \(A\) and \(B\),</div></li><li><div class="list-item-paragraph">\(\Delta OBC\) is similar to \(\Delta TAB\).</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/1768110672714-c5voub3uhv7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgxMTA2NzI3MTQtYzV2b3ViM3VodjcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTYxMiwiZXhwIjoxNzkyNzI1OTEyfQ.AhsQMSWMmRa8mM0ocrOH-DWJJVP4dTexFZaQ0NWt1C0" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the figure, \(TA\) and \(TB\) are tangents to the circle, centre \(O\), from \(T\).

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