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PSLE

PSLE Foundation Mathematics

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<p>The diagram shows a triangle $ABC$ with vertices $A(-5,\ 2)$ and $B(7,\ 8)$. The point $C$ lies on the perpendicular bisector of $AB$ and the equation of the line $BC$ is $y=3x-13$.</p>
<ol class="abc-list"><li data-list-item-id="cg-q13-a" data-list-item-path="cg-q13-a"><div class="list-item-paragraph">Find the equation of the perpendicular bisector of $AB$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q13-b" data-list-item-path="cg-q13-b"><div class="list-item-paragraph">Find the coordinates of $C$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q13-c" data-list-item-path="cg-q13-c"><div class="list-item-paragraph">Find the area of the triangle $ABC$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a triangle $ABC$ with vertices $A(-5,\ 2)$ and $B(7,\ 8)$. The point $C$ lies on the perpendicular bisector of $AB$ and the equation of the line $BC$ is $y=3x-13$.

Find the equation of the perpendicular bisector of $AB$.

Area of Rectilinear Figures

Parallel and Perpendicular Lines

Intersection of Two Lines

Equation of a Straight Line

Midpoint of a Line Segment

Perpendicular Bisectors

Perpendicular bisector and the third vertex of a triangle

Coordinate Geometry

<p>The diagram shows an isosceles triangle $PQR$, where&nbsp;$PQ=QR$ and the line segment $QR$ is parallel to the $y$-axis. Point $P$ lies on the $y$-axis and the equation of the line $PR$ is $x+3y=42$. Point $Q$ is $(-9,\ k)$, where&nbsp;$k$ is a positive constant.</p>
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<ol class="abc-list"><li data-list-item-id="cg-q11-a" data-list-item-path="cg-q11-a"><div class="list-item-paragraph">Find the coordinates of $P$, $Q$ and $R$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q11-b" data-list-item-path="cg-q11-b"><div class="list-item-paragraph">Given that the point $S$ is such that&nbsp;$PQRS$ is a kite and the line segment $RS$ is parallel to the $x$-axis, find the coordinates of $S$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q11-c" data-list-item-path="cg-q11-c"><div class="list-item-paragraph">Find the area of the kite $PQRS$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows an isosceles triangle $PQR$, where $PQ=QR$ and the line segment $QR$ is parallel to the $y$-axis. Point $P$ lies on the $y$-axis and the equation of the line $PR$ is $x+3y=42$. Point $Q$ is $(-9,\ k)$, where $k$ is a positive constant.

Find the coordinates of $P$, $Q$ and $R$.

Given that the point $S$ is such that $PQRS$ is a kite and the line segment $RS$ is parallel to the $x$-axis, find the coordinates of $S$.

Length of a Line Segment

Isosceles triangle extended to a kite

<p>The figure shows a right-angled triangle $ABC$ in which $A$, $B$ and $C$ are $(-3,\ 18)$, $(k,\ 0)$ and $(15,\ 4)$ respectively. $AB$ cuts the $y$-axis at $P$. The point $Q$ lies on $AC$ and $PQ$ is parallel to $BC$.</p>
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<ol class="i-list"><li data-list-item-id="cg-q10-i" data-list-item-path="cg-q10-i"><div class="list-item-paragraph">Given that&nbsp;$k&lt;5$, find the value of $k$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q10-ii" data-list-item-path="cg-q10-ii"><div class="list-item-paragraph">Find the coordinates of $P$ and show that&nbsp;$P$ is the midpoint of $AB$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q10-iii" data-list-item-path="cg-q10-iii"><div class="list-item-paragraph">Hence find the coordinates of $Q$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q10-iv" data-list-item-path="cg-q10-iv"><div class="list-item-paragraph">The point $D$ is such that&nbsp;$ABCD$ is a rectangle. Find the coordinates of $D$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q10-v" data-list-item-path="cg-q10-v"><div class="list-item-paragraph">Find the area of the rectangle $ABCD$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q10-vi" data-list-item-path="cg-q10-vi"><div class="list-item-paragraph">The point $F$ lies on $BC$ such that&nbsp;$BF : FC = 1 : 3$. Find the coordinates of $F$.<span class="inline-marks">[2]</span></div></li></ol>

The figure shows a right-angled triangle $ABC$ in which $A$, $B$ and $C$ are $(-3,\ 18)$, $(k,\ 0)$ and $(15,\ 4)$ respectively. $AB$ cuts the $y$-axis at $P$. The point $Q$ lies on $AC$ and $PQ$ is parallel to $BC$.

Given that $k<5$, find the value of $k$.

Find the coordinates of $P$ and show that $P$ is the midpoint of $AB$.

The point $D$ is such that $ABCD$ is a rectangle. Find the coordinates of $D$.

The point $F$ lies on $BC$ such that $BF : FC = 1 : 3$. Find the coordinates of $F$.

Ratio Theorem and Section Formula

Right-angled triangle completed to a rectangle

<p>The diagram shows a trapezium $ABCD$ in which $A$ is $(1,\ 2)$, $B$ is $(5,\ 10)$ and $D$ is $(9,\ 3)$. Angle $ABC$ is $90^\circ$ and $AB$ is parallel to $DC$.</p>
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<ol class="i-list"><li data-list-item-id="cg-q09-i" data-list-item-path="cg-q09-i"><div class="list-item-paragraph">Find the equation of $CD$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q09-ii" data-list-item-path="cg-q09-ii"><div class="list-item-paragraph">Find the equation of $BC$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q09-iii" data-list-item-path="cg-q09-iii"><div class="list-item-paragraph">Hence find the coordinates of $C$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q09-iv" data-list-item-path="cg-q09-iv"><div class="list-item-paragraph">Find the area of the trapezium $ABCD$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q09-v" data-list-item-path="cg-q09-v"><div class="list-item-paragraph">The point $E$ lies on $AC$ such that&nbsp;$AE : EC = 3 : 2$. Find the coordinates of $E$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a trapezium $ABCD$ in which $A$ is $(1,\ 2)$, $B$ is $(5,\ 10)$ and $D$ is $(9,\ 3)$. Angle $ABC$ is $90^\circ$ and $AB$ is parallel to $DC$.

The point $E$ lies on $AC$ such that $AE : EC = 3 : 2$. Find the coordinates of $E$.

Right-angled trapezium and a point dividing a diagonal

<p>$ABCD$ is a rhombus in which $A$ is $(4,\ 6)$ and $B$ is $(5,\ 14)$. The diagonals $AC$ and $BD$ intersect at $T$, and $AC$ cuts the $y$-axis at $(0,\ 4)$.</p>
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<ol class="i-list"><li data-list-item-id="cg-q07-i" data-list-item-path="cg-q07-i"><div class="list-item-paragraph">Find the equations of $AC$ and $BD$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q07-ii" data-list-item-path="cg-q07-ii"><div class="list-item-paragraph">Calculate the coordinates of $T$, $C$ and $D$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q07-iii" data-list-item-path="cg-q07-iii"><div class="list-item-paragraph">Find the area of the rhombus $ABCD$.<span class="inline-marks">[3]</span></div></li></ol>

$ABCD$ is a rhombus in which $A$ is $(4,\ 6)$ and $B$ is $(5,\ 14)$. The diagonals $AC$ and $BD$ intersect at $T$, and $AC$ cuts the $y$-axis at $(0,\ 4)$.

Calculate the coordinates of $T$, $C$ and $D$.

Diagonals of a rhombus

<p>The diagram shows a kite $ABCD$ in which the side $AB$ is parallel to the $x$-axis. The diagonals $AC$ and $BD$ intersect at $T$.</p>
<div class="graph-block" data-graph-config="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" data-width="440" data-height="250" data-proportional style="width:440px;height:250px;max-width:100%;display:flow-root;margin:0 auto;margin-bottom:0px;--graph-width:440px;--graph-aspect:440 / 250;"></div>
<ol class="i-list"><li data-list-item-id="cg-q06-i" data-list-item-path="cg-q06-i"><div class="list-item-paragraph">It is given that&nbsp;$B$ is $(7,\ 2)$ and the equation of the diagonal $AC$ is $2y=x+7$.</div><ol class="abc-list"><li data-list-item-id="cg-q06-i-a" data-list-item-path="cg-q06-i/cg-q06-i-a"><div class="list-item-paragraph">Find the coordinates of $A$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q06-i-b" data-list-item-path="cg-q06-i/cg-q06-i-b"><div class="list-item-paragraph">Find the equation of the diagonal $BD$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q06-i-c" data-list-item-path="cg-q06-i/cg-q06-i-c"><div class="list-item-paragraph">Hence find the coordinates of $T$ and of $D$.<span class="inline-marks">[4]</span></div></li></ol></li><li data-list-item-id="cg-q06-ii" data-list-item-path="cg-q06-ii"><div class="list-item-paragraph">Given also that the coordinates of $C$ are $(11,\ 9)$, find the area of the kite $ABCD$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a kite $ABCD$ in which the side $AB$ is parallel to the $x$-axis. The diagonals $AC$ and $BD$ intersect at $T$.

It is given that $B$ is $(7,\ 2)$ and the equation of the diagonal $AC$ is $2y=x+7$.

Given also that the coordinates of $C$ are $(11,\ 9)$, find the area of the kite $ABCD$.

Kite with a side parallel to the x-axis

<p>The diagram shows a triangle $ABC$ in which $A$ is $(0,\ 6)$, $B$ is $(a,\ 7)$ and $C$ is $(12,\ 0)$. The point $M$ is the midpoint of $AC$ and $BM$ is perpendicular to $AC$.</p>
<div class="graph-block" data-graph-config="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" data-width="440" data-height="280" style="width:440px;height:280px;max-width:100%;display:block;margin:0 auto;"></div>
<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q05-i" data-list-item-path="cg-q05-i"><div class="list-item-paragraph">the coordinates of $M$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="cg-q05-ii" data-list-item-path="cg-q05-ii"><div class="list-item-paragraph">the equation of the perpendicular bisector of $AC$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q05-iii" data-list-item-path="cg-q05-iii"><div class="list-item-paragraph">the value of $a$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q05-iv" data-list-item-path="cg-q05-iv"><div class="list-item-paragraph">the area of the triangle $ABC$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a triangle $ABC$ in which $A$ is $(0,\ 6)$, $B$ is $(a,\ 7)$ and $C$ is $(12,\ 0)$. The point $M$ is the midpoint of $AC$ and $BM$ is perpendicular to $AC$.

the equation of the perpendicular bisector of $AC$,

Perpendicular bisector of a side of a triangle

<p>The diagram shows a rectangle $ABCD$ in which $A$ and $C$ lie on the $x$-axis. The equation of $AB$ is $3y=x+1$ and the equation of $BD$ is $4y=3x-12$.</p>
<div class="graph-block" data-graph-config="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" data-width="376" data-height="200" data-proportional style="width:376px;height:200px;max-width:100%;display:flow-root;margin:0 auto;margin-bottom:0px;--graph-width:376px;--graph-aspect:376 / 200;"></div>
<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q04-i" data-list-item-path="cg-q04-i"><div class="list-item-paragraph">the coordinates of $B$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q04-ii" data-list-item-path="cg-q04-ii"><div class="list-item-paragraph">the coordinates of the midpoint of $AC$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q04-iii" data-list-item-path="cg-q04-iii"><div class="list-item-paragraph">the coordinates of $A$, $C$ and $D$,<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q04-iv" data-list-item-path="cg-q04-iv"><div class="list-item-paragraph">the length of $BC$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q04-v" data-list-item-path="cg-q04-v"><div class="list-item-paragraph">the area of the rectangle $ABCD$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a rectangle $ABCD$ in which $A$ and $C$ lie on the $x$-axis. The equation of $AB$ is $3y=x+1$ and the equation of $BD$ is $4y=3x-12$.

the coordinates of the midpoint of $AC$,

the coordinates of $A$, $C$ and $D$,

Rectangle with a diagonal along the x-axis

<p>The diagram shows a rectangle $ABCD$ in which $A$ is $(1,\ 2)$ and $D$ is $(5,\ 0)$. The diagonal $AC$ lies on the line $y=x+1$.</p>
<div class="graph-block" data-graph-config="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" data-width="380" data-height="250" data-proportional style="width:380px;height:250px;max-width:100%;display:flow-root;margin:0 auto;margin-bottom:0px;--graph-width:380px;--graph-aspect:380 / 250;"></div>
<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q03-i" data-list-item-path="cg-q03-i"><div class="list-item-paragraph">the equation of $CD$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q03-ii" data-list-item-path="cg-q03-ii"><div class="list-item-paragraph">the coordinates of $C$ and of $B$,<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q03-iii" data-list-item-path="cg-q03-iii"><div class="list-item-paragraph">the length of $AB$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q03-iv" data-list-item-path="cg-q03-iv"><div class="list-item-paragraph">the area of the rectangle $ABCD$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a rectangle $ABCD$ in which $A$ is $(1,\ 2)$ and $D$ is $(5,\ 0)$. The diagonal $AC$ lies on the line $y=x+1$.

Rectangle from a vertex pair and a diagonal

<p>In the diagram, the points $P(-4,\ -3)$, $Q(1,\ 9)$ and $R(13,\ 4)$ form a triangle.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="250" data-proportional style="width:300px;height:250px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:300px;--graph-aspect:300 / 250;"></div>
<ol class="i-list"><li data-list-item-id="cg-q02-i" data-list-item-path="cg-q02-i"><div class="list-item-paragraph">Show that&nbsp;$PQ$ is perpendicular to $QR$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q02-ii" data-list-item-path="cg-q02-ii"><div class="list-item-paragraph">Calculate the area of the triangle $PQR$.<span class="inline-marks">[3]</span></div></li></ol>

In the diagram, the points $P(-4,\ -3)$, $Q(1,\ 9)$ and $R(13,\ 4)$ form a triangle.

Show that $PQ$ is perpendicular to $QR$.

Calculate the area of the triangle $PQR$.

Perpendicular sides and the area of a triangle

<p>The diagram shows a line $DE$ which cuts the $y$-axis at $P$ and a line through $F$ meets the $y$-axis at $Q$ such that angle $FPQ =$ angle $FQP$.</p>
<p>The coordinates of $D$ and $E$ are $(-4, 0)$ and $(10, 7)$ respectively.</p>
<p>$F$ is a point on the line $DE$ such that&nbsp;$DF : FE = 4 : 3$.</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/1782712311122-rmouexzadr.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODI3MTIzMTExMjItcm1vdWV4emFkci5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODU1LCJleHAiOjE3OTI2OTYxNTV9.q5RiyRE-nsdHk-MLe9nvBrfL-A7xQlLYZPthSRO8KmQ" style="max-width: min(380px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Find the coordinates of $F$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Find the equation of the straight line $FQ$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Find the area of the triangle $QFE$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a line $DE$ which cuts the $y$-axis at $P$ and a line through $F$ meets the $y$-axis at $Q$ such that angle $FPQ =$ angle $FQP$.

The coordinates of $D$ and $E$ are $(-4, 0)$ and $(10, 7)$ respectively.

$F$ is a point on the line $DE$ such that $DF : FE = 4 : 3$.

2024 CGSS P2 Q5

<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/1782643154194-60z17dgtnt8.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODI2NDMxNTQxOTQtNjB6MTdkZ3RudDgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NSwiZXhwIjoxNzkyNjk2MTU1fQ.qepPciyj59bHnMIZg_P9ZtJSrE_BFWfwY-68JqgihkU" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The diagram shows a triangle $PQR$ with vertices $P(9,10)$ and $Q(1,-2)$. The point $R$ lies on the perpendicular bisector of $PQ$ and the equation of the line $QR$ is $y = 8x - 10$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Find the equation of the perpendicular bisector of $PQ$.</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Find the coordinates of $R$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">Find the area of triangle $PQR$.</span><span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a triangle $PQR$ with vertices $P(9,10)$ and $Q(1,-2)$. The point $R$ lies on the perpendicular bisector of $PQ$ and the equation of the line $QR$ is $y=8x-10$.

Find the equation of the perpendicular bisector of $PQ$.

2025 DSS PRELIMS P1 Q12

<p>The diagram shows two quadratic curves, $y = -\frac{1}{2}x^2 + 2x$ and $y = -\frac{1}{2}x^2 - x + 3$, which intersect each other at $C$. A straight line touches both quadratic curves at only one point, $A$ and $B$, respectively.<br>Find</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">the equation of the straight line in the form of $y = mx + c$.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">the coordinates of $A$, $B$ and $C$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">the exact area of triangle $ABC$.<span class="inline-marks">[2]</span></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/1770806346981-qa5ypeh48l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA4MDYzNDY5ODEtcWE1eXBlaDQ4bC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODU1LCJleHAiOjE3OTI2OTYxNTV9.prRBlxJitPynLqOBrUpbZmqYMkD9XYD-TxMGyGovWVA" style="max-width: min(270px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows two quadratic curves, $y = -\frac{1}{2}x^2 + 2x$ and $y = -\frac{1}{2}x^2 - x + 3$, which intersect each other at $C$. A straight line touches both quadratic curves at only one point, $A$ and $B$, respectively.
Find

Intersection of a Line and a Curve

Nature of Roots of Quadratic Equations

IP3 Class Test (May 2025) Bonus Question

Quadratic Functions, Equations and Inequalities

<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/1770805603072-58nwvw5dd8k.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA4MDU2MDMwNzItNThud3Z3NWRkOGsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NSwiZXhwIjoxNzkyNjk2MTU1fQ.PCnEB4S6D-TDG0L3mnxKOI2LjNs9ZuNbUKrdBssYXxo" style="max-width: min(250px, 100%); height: auto; display: block;" alt="The diagram shows an isosceles triangle $PQR$ on a coordinate plane." class="aligncenter custom-width" /></figure></div>
<p>The diagram shows an isosceles triangle $PQR$ with vertices at $P(0, k)$, $Q(3, 9)$ and $R(h, 6)$, where&nbsp;$h$ and $k$ are positive constants and $h &gt; k$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$h + k = 12$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">It is now given that&nbsp;$h = 7$.<br>The point $S$ is such that&nbsp;$PQRS$ is a kite. Given that&nbsp;$S$ lies on the line $21y = 7x - 8$, find the coordinates of $S$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Find the area of the kite.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows an isosceles triangle $PQR$ with vertices at $P(0, k)$, $Q(3, 9)$ and $R(h, 6)$, where $h$ and $k$ are positive constants and $h > k$.

It is now given that $h = 7$.\nThe point $S$ is such that $PQRS$ is a kite. Given that $S$ lies on the line $21y = 7x - 8$, find the coordinates of $S$.

IP3 Class Test (May 2025) Q11 (2023 ACSBR AM Prelim P2 Q5)

<p>The vertices of triangle $ABC$ are $A(8,-5)$, $B(11,-4)$ and $C(1,k)$. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $k$ if angle $ACB$ is $90$ degrees,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the area of triangle $ABC$.</span></div></li></ol>

The vertices of triangle $ABC$ are $A(8,-5)$, $B(11,-4)$ and $C(1,k)$. Find

the value of $k$ if angle $ACB$ is $90$ degrees,

<p>Three points have coordinates $P(-3, 4)$, $Q(7, -2)$ and $R(5, m)$. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $m$ if&nbsp;$PQ$ is perpendicular to $PR$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the midpoint of $PQ$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the equation of the perpendicular bisector of $PQ$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the lengths of $PQ$ and $PR$,</span></div></li><li><div class="list-item-paragraph">the area of triangle $PQR$, using the results in <strong>(d)</strong>.</div></li></ol>

Three points have coordinates $P(-3, 4)$, $Q(7, -2)$ and $R(5, m)$. Find

the value of $m$ if $PQ$ is perpendicular to $PR$,

the equation of the perpendicular bisector of $PQ$,

<p>Three points have coordinates $A(1, 8)$, $B(0, 3)$ and $C(k, 5)$. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $k$ if&nbsp;$AC$ is perpendicular to $BC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the midpoint of $AB$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the equation of the perpendicular bisector of $AB$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the lengths of $AB$ and $BC$,</span></div></li><li><div class="list-item-paragraph">the area of triangle $ABC$, using the results in <strong>(d)</strong>.</div></li></ol>

Three points have coordinates $A(1, 8)$, $B(0, 3)$ and $C(k, 5)$. Find

the value of $k$ if $AC$ is perpendicular to $BC$,

the equation of the perpendicular bisector of $AB$,

<p><strong>Solutions to this question by accurate drawing will not be accepted.</strong></p>
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<p>In the diagram above, we have a trapezium $ABCD$.</p>
<p>The line $BC$ has equation $2y + 8x = 6$ and $D$ has coordinates $(-4, 0)$. $B$ cuts the $y$-axis.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Given that&nbsp;$DC$ is perpendicular to $BC$, find $C$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find $A$.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Calculate the area of $ABCD$.</div></li></ol>

**Solutions to this question by accurate drawing will not be accepted.**

In the diagram above, we have a trapezium $ABCD$.

The line $BC$ has equation $2y + 8x = 6$ and $D$ has coordinates $(-4, 0)$. $B$ cuts the $y$-axis.

Given that $DC$ is perpendicular to $BC$, find $C$.

<p><strong>Solutions to this question by accurate drawing will not be accepted.</strong></p>
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<p>The diagram shows a rhombus $ABCD$ with vertices $A$ and $C$ at the points $(0, 2)$ and $(-6, 4)$ respectively. $B$ lies on the $x$-axis and the line $DC$ produced intersects with the $x$-axis at $E$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the coordinates of $B$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Explain if&nbsp;$ABCD$ is a square.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the coordinates of $E$.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find the area of $ABED$.</div></li></ol>

The diagram shows a rhombus $ABCD$ with vertices $A$ and $C$ at the points $(0, 2)$ and $(-6, 4)$ respectively. $B$ lies on the $x$-axis and the line $DC$ produced intersects with the $x$-axis at $E$.

<p><strong>Solutions to this question by accurate drawing will not be accepted.</strong></p>
<p>In the diagram, $ABCD$ is a trapezium where&nbsp;$AD$ and $BC$ are parallel to the $y$-axis and $AB$ is parallel to the $x$-axis. $D$ is the point $(-4, 7)$ and $C$ is the point $(5, 9)$. The line $DC$ cuts the $y$-axis at $Q$.</p>
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<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the equation of the line $CD$,</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the coordinates of $Q$,</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Given the area of the trapezium is $162$ units$^2$, find the coordinates of point $B$,</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Hence, find the equation of the line $AC$.</div></li></ol>

In the diagram, $ABCD$ is a trapezium where $AD$ and $BC$ are parallel to the $y$-axis and $AB$ is parallel to the $x$-axis. $D$ is the point $(-4, 7)$ and $C$ is the point $(5, 9)$. The line $DC$ cuts the $y$-axis at $Q$.

Given the area of the trapezium is $162$ units$^2$, find the coordinates of point $B$,

Hence, find the equation of the line $AC$.

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