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<p>In the diagram, $ABCD$ and $DEFG$ are squares. Prove that</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1770110251284-1otu4hufsre.png" style="max-width: min(160px, 100%); height: auto; display: block;" alt="Geometric diagram showing two squares sharing vertex D" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-content"><span class="list-item-text">triangles $ADE$ and $CDG$ are congruent, and</span></div></li><li data-list-item-id="item-ii"><div class="list-item-content"><span class="list-item-text">$AE$ is perpendicular to $CG$.</span></div></li></ol>

In the diagram, $ABCD$ and $DEFG$ are squares. Prove that

triangles $ADE$ and $CDG$ are congruent, and

Congruence and Similar Figures

<p>A building has a perimeter of $8$ cm, with a gate of width $0.5$ cm on a map. If the actual perimeter of the building is $0.56$ km,</p>
<ol class="abc-list"><li><div class="list-item-paragraph">find the scale of the map in the form $1:r$.</div></li><li><div class="list-item-paragraph">A model of the building is made to another scale of $1:5000$.</div><div class="list-item-paragraph">Calculate</div><ol class="i-list"><li><div class="list-item-paragraph">the width of the gate in centimetres as represented by the second scale.</div></li><li><div class="list-item-paragraph">the floor area of the building model in square centimetres if the actual floor area of the building is $16000$ m$^{2}$.</div></li></ol></li></ol>

A building has a perimeter of $8$ cm, with a gate of width $0.5$ cm on a map. If the actual perimeter of the building is $0.56$ km,

Scale Diagrams and Building Model Calculations

<p>The Circle Line is the fourth MRT line in Singapore. This underground line is $35.7$ km long. It is represented by $7.14$ cm on Map $A$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express the map scale in the form $1:n$.</div></li><li><div class="list-item-paragraph">Downtown Line, the fifth MRT line, is represented by $0.86$ cm on the same map. </div><div class="list-item-paragraph">Find its length on another map with the scale of $1:20\,000$.</div></li><li><div class="list-item-paragraph">Kim Chuan SMRT Depot is an underground depot that houses the trains for the Circle Line and Downtown Line. It has an area of $2.5$ cm$^{2}$ on the map $B$. What is the actual area of the depot in km$^{2}$?</div></li></ol>

The Circle Line is the fourth MRT line in Singapore. This underground line is $35.7$ km long. It is represented by $7.14$ cm on Map $A$.

Map Scale Calculations Involving Distance and Area

<p>The actual area of Sungei Buloh Wetland Reserve is $1.3$ km$^{2}$. A scale of $1$ cm represents $1.5$ km on the map.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express the scale in the form $1:n$.</div></li><li><div class="list-item-paragraph">Calculate the area of Sungei Buloh Wetland Reserve on the map in cm$^{2}$.</div></li></ol>

The actual area of Sungei Buloh Wetland Reserve is $1.3$ km$^{2}$. A scale of $1$ cm represents $1.5$ km on the map.

Map Scale and Area Conversion Problem

<p>The area of a camp, $27$ km$^{2}$, is represented on a map by $3$ cm$^{2}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">The map has a scale of $1:r$. Find the value of $r$.</div></li><li><div class="list-item-paragraph">The perimeter of the camp is $13$ km. Find the perimeter of the camp on the map.</div></li></ol>

The area of a camp, $27$ km$^{2}$, is represented on a map by $3$ cm$^{2}$.

Map Scale Problem Involving Area and Perimeter

<p>A map is drawn to a scale of $1:125\,000$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the actual distance, in kilometres, represented by $7$cm on the map.</div></li><li><div class="list-item-paragraph">A new housing development in Yishun has an area of $0.8$km$^{2}.$ Find the area represented by the housing development on the map.</div></li></ol>

A map is drawn to a scale of $1:125\,000$.

Map Interpretation: Linear and Area Scale Calculations

<p>Trapezium $WXYZ$ is an enlargement of trapezium $ABCD$. The lengths shown in figure below are in centimetres.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1764923064634-3ne1r39pu1z.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the scale factor of the enlargement.</div></li><li><div class="list-item-paragraph">Find $\angle BCD$, stating your reason clearly.</div></li><li><div class="list-item-paragraph">Given that the area of trapezium $WXYZ$ is $124.5$cm$^{2}$. Find the height of the trapezium $WXYZ$.</div></li></ol>

Trapezium $WXYZ$ is an enlargement of trapezium $ABCD$. The lengths shown in figure below are in centimetres.

Trapezium Similarity: Side Ratios, Angles and Area

<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;$\Delta ABE$ is similar to triangle $ACD$.</p><ol class="abc-list"><li><div class="list-item-paragraph">Calculate the length of $CD$.</div></li><li><div class="list-item-paragraph">Find the scale factor enlarging $\Delta ABE$ to $\Delta ACD$.</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/public/question-images/pasted/1764922768644-670rm26vk3p.png" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

Similar Triangles and Scale Factor in $\Delta ABE$  and $\Delta ACD$ 

<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>Triangles $ABC$ and $AED$ in the diagram above are similar.</p><p>Find</p><ol class="abc-list"><li><div class="list-item-paragraph">$a$,</div></li><li><div class="list-item-paragraph">$x$.</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/public/question-images/pasted/1764922459351-v8hf90d3dhs.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

Similar Triangles $𝐴𝐵𝐶$ and $𝐴𝐸𝐷$: Finding Angle and Side Length

<p>In the diagram below, triangle $ABX$ is similar to triangle $CDX$.</p>
<p>Given that&nbsp;$AB=3$ cm, $AX=2$cm and $CX=7$ cm, 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 length of $CD$,</div></li><li><div class="list-item-paragraph">the ratio of $DX:DB$.</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/public/question-images/pasted/1773664783950-er8urqa6ml.png" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the diagram below, triangle $ABX$ is similar to triangle $CDX$.

Given that $AB=3$ cm, $AX=2$cm and $CX=7$ cm, find

Similar Triangles in an Intersecting Segments Diagram

<p>In the figure below, rectangle $ABCD$ is reduced to rectangle $EFGH$.</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>From the information above, find the value of $p$.</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/public/question-images/pasted/1764921999385-t2sjpainffl.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the figure below, rectangle $ABCD$ is reduced to rectangle $EFGH$.

Reduction of Rectangles: Finding the Scale and Unknown Length

<p>The diagram below shows an irregular pentagon $PQRST$ which is an enlargement of the irregular pentagon $ABCDE$. 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">$\angle RST$,</div></li><li><div class="list-item-paragraph">the scale factor of the enlargement.</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/public/question-images/pasted/1764921662745-7h5cbrg947t.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The diagram below shows an irregular pentagon $PQRST$ which is an enlargement of the irregular pentagon $ABCDE$. Find

Enlargement of an Irregular Pentagon: Angle and Scale Factor

<p>Two congruent quadrilaterals, $ABCD$ and $WXYZ$, are given below (not drawn to scale).</p>
<p>$\angle BCD=\angle XYZ=90{}^\circ $, $\angle ABC=9x{}^\circ $ and $\angle ACD=2x{}^\circ $. $XWV$ is a straight line such that&nbsp;$\angle VWZ=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;"><ol class="abc-list"><li><div class="list-item-paragraph">Find $\angle XWZ$.</div></li><li><div class="list-item-paragraph">Using your result in <strong>(a)</strong>, find the value of $x$.</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/public/question-images/pasted/1764921354600-59odscdve7s.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

Two congruent quadrilaterals, $ABCD$ and $WXYZ$, are given below (not drawn to scale).

$\angle BCD=\angle XYZ=90{}^\circ $, $\angle ABC=9x{}^\circ $ and $\angle ACD=2x{}^\circ $. $XWV$ is a straight line such that $\angle VWZ=42{}^\circ $.

Congruent Quadrilaterals and Exterior Angles

<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, $ABCD$ is congruent to $WXYZ$, find the</p><ol class="abc-list"><li><div class="list-item-paragraph">length of $XW$,</div></li><li><div class="list-item-paragraph">value of $y{}^\circ $,</div></li><li><div class="list-item-paragraph">value of $x{}^\circ $.</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/public/question-images/pasted/1764921020039-40yvkky6bzt.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

Congruent Quadrilaterals: Finding Missing Side and Angles

<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>Triangle $ABC$ is congruent to triangle $XYZ$</p><p>Find</p><ol class="abc-list"><li><div class="list-item-paragraph">angle $x$,</div></li><li><div class="list-item-paragraph">angle $z$,</div></li><li><div class="list-item-paragraph">$XZ$.</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/public/question-images/pasted/1764919997670-hg6e0u4xsbe.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

Congruent Triangles: Finding Angles and Side Length

<p>An area of $4$ cm$^{2}$ on a map represents an actual area of $2.25$ km$^{2}$.</p>
<p>Calculate</p>
<ol class="abc-list"><li><div class="list-item-paragraph">the area on the map, in square centimetres, which represents an actual area of $27$ km$^{2}$.</div></li><li><div class="list-item-paragraph">the actual distance, in kilometres, represented by a length of $7.5$ cm.</div></li><li><div class="list-item-paragraph">the scale of the map, in the form $1:n$.</div></li></ol>

An area of $4$ cm$^{2}$ on a map represents an actual area of $2.25$ km$^{2}$.

Map Scales

Map Scale: Area and Distance Conversions

<p>A map is drawn to a scale of $1:25\,000$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">This scale represents $1$ cm$ : n $ km. Find $n$.</div></li><li><div class="list-item-paragraph">The distance between two towns on the map is $30$ cm. Find the actual distance, in kilometres, between the two towns.</div></li><li><div class="list-item-paragraph">A lake has an actual area of $2.5$ km$^{2}$. Find the area, in square centimetres, of the lake on the map.</div></li></ol>

A map is drawn to a scale of $1:25\,000$.

Map Scale Calculations: Distance and Area

<p>A map is drawn to the scale of $1:30\,000$. A road is represented by a line of $3.2$cm on the map. Calculate the actual length of the road, giving your answers in meters.</p>

A map is drawn to the scale of $1:30\,000$. A road is represented by a line of $3.2$cm on the map. Calculate the actual length of the road, giving your answers in meters.

Map Scale Problem: Determining Actual Distance

<p>An area of $4$ cm$^{2}$ on a map represents an actual area of $0.36$ km$^{2}$.</p>
<p>Calculate</p>
<ol class="abc-list"><li><div class="list-item-paragraph">the actual area in square kilometres represented by $80$ cm$^{2}$ on the map,</div></li><li><div class="list-item-paragraph">the scale of the map in the form $1:n$,</div></li><li><div class="list-item-paragraph">the distance on the map in centimetres which represents an actual distance of $6$ km.</div></li></ol>

An area of $4$ cm$^{2}$ on a map represents an actual area of $0.36$ km$^{2}$.

Area and Distance Calculations from a Map Scale

<p>A map has a scale of $1:400\,000$. Given that the </p>
<ol class="abc-list"><li><div class="list-item-paragraph">actual distance between two locations $A$ and $B$ is $14$ km, find the distance, in cm, between $A$ and $B$ on the map.</div></li><li><div class="list-item-paragraph">area of a garden on the map is $1.5$ cm$^{2}$, calculate its actual area in km$^{2}$.</div></li></ol>

A map has a scale of $1:400\,000$. Given that the 

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