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<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="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjM4MCwiaGVpZ2h0Ijo0MjAsImVsZW1lbnRzIjpbeyJpZCI6ImNnLXEwMy1zMSIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbMSwyXSwicG9pbnQyIjpbNywxNF0sInN0cm9rZVdpZHRoIjoxLjR9LHsiaWQiOiJjZy1xMDMtczIiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzcsMTRdLCJwb2ludDIiOlsxMSwxMl0sInN0cm9rZVdpZHRoIjoxLjR9LHsiaWQiOiJjZy1xMDMtczMiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzExLDEyXSwicG9pbnQyIjpbNSwwXSwic3Ryb2tlV2lkdGgiOjEuNH0seyJpZCI6ImNnLXEwMy1zNCIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbNSwwXSwicG9pbnQyIjpbMSwyXSwic3Ryb2tlV2lkdGgiOjEuNH0seyJ4IjoxLjUsInkiOjIuNSwiaWQiOiJjZy1xMDMtbGEiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoQSgxLFxcIDIpXFwpIn0seyJ4Ijo3LCJ5IjoxNC41LCJpZCI6ImNnLXEwMy1sYiIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXChCXFwpIn0seyJ4IjoxMS41LCJ5IjoxMiwiaWQiOiJjZy1xMDMtbGMiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoQ1xcKSJ9LHsieCI6NS40LCJ5IjotMC45LCJpZCI6ImNnLXEwMy1sZCIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXChEKDUsXFwgMClcXCkifV0sInhBeGlzTGFiZWwiOiJcXCh4XFwpIiwieUF4aXNMYWJlbCI6IlxcKHlcXCkiLCJib3VuZGluZ0JveCI6Wy0zLDE4LDE2LC0zXSwib3JpZ2luTGFiZWwiOiJcXChPXFwpIiwic2hvd0F4aXNMYWJlbHMiOmZhbHNlLCJzaG93TmF2aWdhdGlvbiI6ZmFsc2UsImtlZXBBc3BlY3RSYXRpbyI6dHJ1ZX0=" data-width="380" data-height="420" style="width:380px;height:420px;max-width:100%;display:block;margin:0 auto;"></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 $y=x+1$.

Length of a Line Segment

Area of Rectilinear Figures

Parallel and Perpendicular Lines

Equation of a Straight Line

Midpoint of a Line Segment

Rectangle from a vertex pair and a diagonal

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>
<div class="graph-block" data-graph-config="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" data-width="340" data-height="408" style="width:340px;height:408px;max-width:100%;display:block;margin:0 auto;"></div>
<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 $S$.

Isosceles triangle extended to a kite

<p>The diagram shows the points $A(-7,\ 5)$ and $B(5,\ -4)$.</p>
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<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q01-i" data-list-item-path="cg-q01-i"><div class="list-item-paragraph">the length of $AB$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q01-ii" data-list-item-path="cg-q01-ii"><div class="list-item-paragraph">the coordinates of the midpoint of $AB$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="cg-q01-iii" data-list-item-path="cg-q01-iii"><div class="list-item-paragraph">the gradient of $AB$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="cg-q01-iv" data-list-item-path="cg-q01-iv"><div class="list-item-paragraph">the equation of $AB$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows the points $A(-7,\ 5)$ and $B(5,\ -4)$.

the coordinates of the midpoint of $AB$,

Gradient of a Straight Line

Length, midpoint, gradient and equation from two points

<p>A circle has the equation $x^2 + y^2 - 8x - 4y = 5$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the radius of the circle and the coordinates of its centre.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The line $y = x - 1$ intersects the circle at two points $M$ and $N$.</div><div class="list-item-paragraph">Show that the coordinates of $M$ and $N$ are $(0, -1)$ and $(7, 6)$ respectively.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Determine if $MN$ is the diameter of the circle.</div><div class="list-item-paragraph">Show your working clearly.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the equation of the tangent of the circle at point $N$.</span><span class="inline-marks">[3]</span></div></li></ol>

A circle has the equation $x^2 + y^2 - 8x - 4y = 5$.

Find the radius of the circle and the coordinates of its centre.

Find the equation of the tangent of the circle at point $N$.

2023 CSS AM S3 EOY Q11 

<p><strong>Solutions to this question by accurate drawing will not be accepted.</strong><br>$A$, $B$ and $C$ are the points $\left( -10,15 \right)$, $\left( -10,-10 \right)$ and $\left( 10,-20 \right)$.</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 data-list-item-id="item-1773733620577-f2bplw13j"><div class="list-item-paragraph">Find the area of triangle $ABC$.</div></li><li data-list-item-id="item-1773733621100-6wlziru6j"><div class="list-item-paragraph">$ABCD$ is a parallelogram. Find the coordinates of the point $D$.</div></li><li data-list-item-id="item-1773733621620-mlysy8cor"><div class="list-item-paragraph">$ABCE$ is a trapezium with $AB$ parallel to $CE$. The area of the trapezium is $650$ units$^{2}$.<br>Find the coordinates of the point $E$.</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/converted/1760169362834-kz0oiwizouf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAxNjkzNjI4MzQta3owb2l3aXpvdWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTYyMjkzNSwiZXhwIjoxNzg5NjIzMjM1fQ.M6O-FR7LPH3cCrXOxYKLa04BIgYp9gcbV67Z_3wGlTo" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

**Solutions to this question by accurate drawing will not be accepted.**
$A$, $B$ and $C$ are the points $\left( -10,15 \right)$, $\left( -10,-10 \right)$ and $\left( 10,-20 \right)$.

Quadratic Functions - General

Advanced coordinate geometry: Using Coordinate Geometry to Find Missing Points

Quadratic Functions, Equations and Inequalities

<p>Find the equation of a circle whose centre is $\left( 5,-2 \right)$ and which touches the $y$-axis.</p>

Find the equation of a circle whose centre is $\left( 5,-2 \right)$ and which touches the $y$-axis.

Further Coordinate Geometry — General

Finding equation of circle, given center and after deducing a point on the circle

<p>Find the area of polygon whose vertices are $\left( 0,-3 \right)$, $\left( -1,5 \right)$, $\left( -4,2 \right)$, $\left( 1,6 \right)$, $\left( -1,-4 \right)$ and $\left( 5,1 \right)$.</p>

Find the area of polygon whose vertices are $\left( 0,-3 \right)$, $\left( -1,5 \right)$, $\left( -4,2 \right)$, $\left( 1,6 \right)$, $\left( -1,-4 \right)$ and $\left( 5,1 \right)$.

Area of Polygon Using the Shoelace Formula

<p><strong>Solutions to this question by accurate drawing will not be accepted. </strong><br>The diagram shows a quadrilateral $ABCD$ in which $A$ is $\left( 2,5 \right)$, $B$ is $\left( 4,-1 \right)$ and point $C$ is on the $y$-axis.</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 data-list-item-id="item-1773734128699-ipkw3thyw"><div class="list-item-paragraph">Find the gradient of the line perpendicular to $AB$.</div></li><li data-list-item-id="item-1773734129142-z5nzfzriw"><div class="list-item-paragraph">If the perpendicular bisector of $AB$ cuts the $y$-axis at $C$, find the coordinates of $C$.</div></li><li data-list-item-id="item-1773734129638-39pzygoqn"><div class="list-item-paragraph">Find the coordinates of $D$ if&nbsp;$ABCD$ is a parallelogram.</div></li><li data-list-item-id="item-1773734130206-ftqh23kre"><div class="list-item-paragraph">Calculate the area of the parallelogram $ABCD$.</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/converted/1760171013454-sawp8q35xf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAxNzEwMTM0NTQtc2F3cDhxMzV4Zi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NjIyOTM1LCJleHAiOjE3ODk2MjMyMzV9.hrug6rI1z3o3VnUHJvO3PrhJJ3uk1C6ers6hKEeY8EI" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

**Solutions to this question by accurate drawing will not be accepted. **
The diagram shows a quadrilateral $ABCD$ in which $A$ is $\left( 2,5 \right)$, $B$ is $\left( 4,-1 \right)$ and point $C$ is on the $y$-axis.

Advanced coordinate geometry: Determining Area from Coordinates

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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 $AB$.

Perpendicular bisector and the third vertex of a triangle

<p>Find the coordinates of the midpoint of the line segment joining</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="i-list"><li data-list-item-id="item-1780384397821-xhsom5oj4"><div class="list-item-paragraph">$A\left( 4,4 \right)$ and $B\left( 7,3 \right)$</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="2"><li data-list-item-id="item-1780384405026-q7ghq1a25"><div class="list-item-paragraph">$C\left( 3,-2 \right)$ and $D\left( 0,-\frac{1}{2} \right)$</div></li></ol></div></div></div>

Find the coordinates of the midpoint of the line segment joining

Calculating Midpoints on Cartesian Plane

<p>In the diagram, the points $P(-4,\ -3)$, $Q(1,\ 9)$ and $R(13,\ 4)$ form a triangle.</p>
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<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 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>
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<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>If $A\left( 1,1 \right)$, $B(0,6)$ and $C\left( -2,7 \right)$ are three vertices of a parallelogram $ABCD$, find the midpoint of $AC$, find the coordinates of $D$.</p>

If $A\left( 1,1 \right)$, $B(0,6)$ and $C\left( -2,7 \right)$ are three vertices of a parallelogram $ABCD$, find the midpoint of $AC$, find the coordinates of $D$.

Analytic Geometry: Linear Bisection and Midpoints

<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>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>
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<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 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 $F$.

Right-angled triangle completed to a rectangle

<p>Find the equation of the circle with centre $C(-3,8)$ and a point on the circle of $\left( 2,-4 \right)$.</p>

Find the equation of the circle with centre $C(-3,8)$ and a point on the circle of $\left( 2,-4 \right)$.

<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="340" style="width:440px;height:340px;max-width:100%;display:block;margin:0 auto;"></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>Find the equation of a circle with diameter $AB$, where coordinates of $A$ and $B$ are $\left( 4,-3 \right)$ and $\left( 2,1 \right)$ respectively.</p>

Find the equation of a circle with diameter $AB$, where coordinates of $A$ and $B$ are $\left( 4,-3 \right)$ and $\left( 2,1 \right)$ respectively.

Math Question Bank