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PSLE Foundation Mathematics

<p>The diagram shows a coastal reserve bounded by four survey markers $P$, $Q$, $R$ and $S$ on level ground.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<p>The bearing of $P$ from $S$ is $342^\circ$ and $SP=176\text{ m}$. Marker $Q$ is $80\text{ m}$ due west of $P$.</p>
<p>The angle $PQR$ is $117^\circ$ and the bearing of $P$ from $R$ is $052^\circ$.</p>
<p>Give non-exact numerical answers correct to $3$ significant figures unless a different degree of accuracy is stated.</p>
<ol class="abc-list"><li data-list-item-id="emath-aot-q-a" data-list-item-path="emath-aot-q-a"><div class="list-item-paragraph">Find the bearing of $R$ from $P$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="emath-aot-q-b" data-list-item-path="emath-aot-q-b"><div class="list-item-paragraph">Find $\angle QPR$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="emath-aot-q-c" data-list-item-path="emath-aot-q-c"><div class="list-item-paragraph">Show that the distance $PR$ is $168.66\text{ m}$, correct to two decimal places.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q-d" data-list-item-path="emath-aot-q-d"><div class="list-item-paragraph">Find the area of triangle $PQR$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="emath-aot-q-e" data-list-item-path="emath-aot-q-e"><div class="list-item-paragraph">Find $\angle RPS$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="emath-aot-q-f" data-list-item-path="emath-aot-q-f"><div class="list-item-paragraph">Hence, calculate the distance $SR$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q-g" data-list-item-path="emath-aot-q-g"><div class="list-item-paragraph">An observation tower, $24\text{ m}$ high, stands vertically at $P$.</div><div class="list-item-paragraph">Find the angle of elevation of the top of the tower from $S$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a coastal reserve bounded by four survey markers $P$, $Q$, $R$ and $S$ on level ground.

The bearing of $P$ from $S$ is $342^\circ$ and $SP=176\text{ m}$. Marker $Q$ is $80\text{ m}$ due west of $P$.

The angle $PQR$ is $117^\circ$ and the bearing of $P$ from $R$ is $052^\circ$.

Give non-exact numerical answers correct to $3$ significant figures unless a different degree of accuracy is stated.

Area of Triangle

Sine Rule

Angles of Elevation and Depression

Cosine Rule

Bearings

Surveying a coastal reserve

Trigonometry

<p>In the diagram, $P$, $Q$ and $R$ represent three reservoirs.</p>
<p>$P$ is $22$ km from $Q$ on a bearing of $140{}^\circ$ and $R$ is $20$ km from $P$ on a bearing of $290{}^\circ$.</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/1767444241225-yqhegew0xe.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0NDQyNDEyMjUteXFoZWdldzB4ZS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.2FzFsNLhUd2P7msBRBy-hdVdueNQop9D_jwswycZCWI" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Calculate the bearing of $Q$ from $P$.</span></div></li><li><div class="list-item-paragraph">A cyclist leaves $P$ and cycled along $PQ$ towards $Q$ at a steady speed of $10$ km/h.</div><div class="list-item-paragraph">He stops at a point $S$ which is due east of $R$.</div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Mark the point $S$ on the diagram.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Calculate the distance $PS$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Calculate the time taken to reach $S$ from $P$, giving your answer in minutes.</span></div></li></ol></li><li><div class="list-item-paragraph">A tower of height $60$ m is located at $P$.</div><div class="list-item-paragraph">What is the angle of elevation of the top of the tower from the cyclist at point $S$?</div></li></ol>

In the diagram, $P$, $Q$ and $R$ represent three reservoirs.

$P$ is $22$ km from $Q$ on a bearing of $140{}^\circ$ and $R$ is $20$ km from $P$ on a bearing of $290{}^\circ$.

Calculate the bearing of $Q$ from $P$.

Speed, Distance and Time

Navigational Trigonometry: Analyzing Bearings and Positions of Three Reservoirs

Ratio, Rate and Speed

<p>In the diagram below (not drawn to scale), $E$, $G$ and $C$ represent three points.</p>
<p>Given that&nbsp;$EG=1125$ m and $CE=957$ m, the bearing of $G$ from $E$ is $135{}^\circ$ and $\angle CEG=60{}^\circ$.</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/1767443723956-904x6d4dtde.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0NDM3MjM5NTYtOTA0eDZkNGR0ZGUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDczMywiZXhwIjoxNzkzNTA1MDMzfQ.2ENzJY8MiZ6Q2qnkt5xMQ_imPxoSDwDqcMYdMxgWCUQ" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the bearing of $E$ from $C$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the distance between $C$ and $G$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\angle EGC$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the bearing of $C$ from $G$.</span></div></li></ol>

In the diagram below (not drawn to scale), $E$, $G$ and $C$ represent three points.

Given that $EG=1125$ m and $CE=957$ m, the bearing of $G$ from $E$ is $135{}^\circ$ and $\angle CEG=60{}^\circ$.

3D Trigonometry: Geometric Analysis of a Right-Angled Triangular Prism

<p>The diagram shows three towns, $A$, $B$ and $C$. $C$ is equidistant from $A$ and $B$.</p>
<p>The bearing of $C$ from $A$ is $135{}^\circ$ and $\angle BAC=65{}^\circ$.</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/1767443330982-qeh0fxhswv.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0NDMzMzA5ODItcWVoMGZ4aHN3di5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.tRKIyF4VAsIMJM1xP6NbxdxFLvjcE6if6fvBouDBGec" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Calculate the</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Reflex angle $ACB$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">bearing of $A$ from $C$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">bearing of $A$ from $B$.</span></div></li></ol>

The diagram shows three towns, $A$, $B$ and $C$. $C$ is equidistant from $A$ and $B$.

The bearing of $C$ from $A$ is $135{}^\circ$ and $\angle BAC=65{}^\circ$.

Navigational Trigonometry: Analyzing Bearings and Positions of Three Towns

<p>$A$, $B$, $C$ and $D$ are four points at sea level.<br>$A$ is due west of $B$, $AB=320$ m, $AD=132$ m, $BC=180$ m, $\angle ADC=48{}^\circ$.<br>The bearing of $B$ from $C$ is $160{}^\circ$.</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">the bearing of $C$ from $B$,</div></li><li><div class="list-item-paragraph">$\angle ABC$,</div></li><li><div class="list-item-paragraph">distance $CA$,</div></li><li><div class="list-item-paragraph">$\angle ACD$,</div></li><li><div class="list-item-paragraph">area of $\Delta ABC$.</div></li></ol></li><li><div class="list-item-paragraph">A boy stands on a vertical tower of height $35$ m at $C$ and looks at a boat travelling along $AB$. Find</div><ol class="i-list"><li><div class="list-item-paragraph">the shortest distance of $C$ to the boat.</div></li><li><div class="list-item-paragraph">the greatest angle of depression from $C$ to the boat.</div></li></ol></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/1767439445116-qbqce3n91s9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0Mzk0NDUxMTYtcWJxY2UzbjkxczkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDczMywiZXhwIjoxNzkzNTA1MDMzfQ.I5thf_E9U3BWCAT8DyXsRkY5Q9HkDyHgtlkwtgviCnA" style="max-width: min(280px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

$A$, $B$, $C$ and $D$ are four points at sea level.
$A$ is due west of $B$, $AB=320$ m, $AD=132$ m, $BC=180$ m, $\angle ADC=48{}^\circ$.
The bearing of $B$ from $C$ is $160{}^\circ$.

Trigonometry in Two and Three Dimensions

3-Dimensional Problem

<p>The diagram (not drawn to scale) represents four points $A$, $B$, $C$ and $D$ on level ground.<br>$ABC$ is a straight line with $D$ due east of $A$ and $\angle ABD=112{}^\circ$.<br>It is also given that&nbsp;$AB=850$ m, $BD=440$ m and $CD=630$ m.<br>Calculate</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><div class="list-item-paragraph">the length of $AD$,</div></li><li><div class="list-item-paragraph">$\angle BCD$,</div></li><li><div class="list-item-paragraph">the shortest distance from $D$ to $BC$,</div></li><li><div class="list-item-paragraph">the bearing of $A$ from $C$.</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/1767434583108-kallh261sz.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MzQ1ODMxMDgta2FsbGgyNjFzei5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.6GRA6RAlzuA2nt-wclbNr0WmuVqtQqpPMyG89pxLAzg" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The diagram (not drawn to scale) represents four points $A$, $B$, $C$ and $D$ on level ground.
$ABC$ is a straight line with $D$ due east of $A$ and $\angle ABD=112{}^\circ$.
It is also given that $AB=850$ m, $BD=440$ m and $CD=630$ m.
Calculate

The Port $P$ Problem: Tracking Ships $Q$ and $R$

<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;"><p>Two ships $Q$ and $R$ are on the bearings of $035{}^\circ$ and $115{}^\circ$ from a port $P$ respectively.<br>The distance between $Q$ and $P$ is $500$ m and the distance between $R$ and $P$ is $900$ m.<br>Calculate</p><ol class="i-list"><li><div class="list-item-paragraph">$\angle QPR$.</div></li><li><div class="list-item-paragraph">the distance $QR$, giving your answer to the nearest m.</div></li><li><div class="list-item-paragraph">the bearing of $Q$ from $R$.</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/1767433802643-u66fwiwmbv.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MzM4MDI2NDMtdTY2Zndpd21idi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.GxZHuLvGDVTsO7Q060e_6o0kcRorQTsS_O4lMfT2m7g" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

Navigating the Open Sea: A Study of Ships $Q$ and $R$

<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 bearing of $D$ from $A$ is $030{}^\circ$, $\angle ABC=50{}^\circ$ and $AB=BC$.<br>Find</p><ol class="i-list"><li><div class="list-item-paragraph">the bearing of $A$ from $D$.</div></li><li><div class="list-item-paragraph">the bearing of $C$ from $B$.</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/1767433367699-fwnj5dnm97i.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MzMzNjc2OTktZnduajVkbm05N2kucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDczMywiZXhwIjoxNzkzNTA1MDMzfQ.do9aGf0zfgxqGdlX2ZI2paEGzImxTAygJ2q_8Or3jcA" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

Calculating Bearings from North

<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>$A$, $B$ and $C$ are three points on level ground and $AC=BC$.<br>$B$ is on a bearing of $075{}^\circ$ from $A$.<br>$C$ is on a bearing of $114{}^\circ$ from $A$.<br>Find</p><ol class="i-list"><li><div class="list-item-paragraph">$\angle ABC$,</div></li><li><div class="list-item-paragraph">the bearing of $C$ from $B$,</div></li><li><div class="list-item-paragraph">the bearing of $A$ from $C$.</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/1767432796044-sw52u7jcfs.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MzI3OTYwNDQtc3c1MnU3amNmcy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.RnzVIelE9cxrVVDWVrTkTlKzKD6KxTHLYVNgvrNL7bM" style="max-width: min(180px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

Navigating by Numbers: The Ultimate Bearings Guide

<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;"><p>$D$, $E$ and $F$ are three points on level ground.<br>Given that&nbsp;$DE=DF$, $\angle FDE=94{}^\circ$ and the bearing of $D$ from $F$ is $032{}^\circ$.<br>Calculate the bearing of</p><ol class="i-list"><li><div class="list-item-paragraph">$E$ from $F$.</div></li><li><div class="list-item-paragraph">$E$ from $D$.</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/1767432232186-kwhr3jfgqc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MzIyMzIxODYta3docjNqZmdxYy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.i83-X3hxcEO9SSCAgrtifkvqJt_qtI2lzx8TK3o_1-s" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

True North: Mastering Three-Figure Bearings

<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 style="text-align: left">In the diagram below, the bearing of $R$ from $P$ is $100{}^\circ$.<br>Given that&nbsp;$\angle PQR=105{}^\circ$, $PR=50$ m and $RQ=28$ m.<br>Find the bearing of $Q$ from $P$.</p></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/1767431928995-mwok2icwm7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MzE5Mjg5OTUtbXdvazJpY3dtNy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.L8HQZ60F-qcBU3C666iSxOyd1wjTRUgntCfm5yTwa2o" style="max-width: min(230px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

Navigating the North: Bearing Basics

<p></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;"><p>Two ships $A$ and $B$ leave a port $P$ at $0915$ hour.<br>Ship $A$ sails on a bearing of $120{}^\circ$ at $25$ km/h for an hour and stops at $X$.<br>Ship $B$ sails on a bearing of $160{}^\circ$ at $40$ km/h for an hour and stops at $Y$.</p><ol class="abc-list"><li><div class="list-item-paragraph">After an hour, find the</div><ol class="i-list"><li><div class="list-item-paragraph">distance between ships $A$ and $B$.</div></li><li><div class="list-item-paragraph">bearing of ship $A$ from ship $B$.</div></li><li><div class="list-item-paragraph">Find how far is ship $B$ due east of port $P$.</div></li></ol></li><li><div class="list-item-paragraph">If ship $B$ anchored at point $Y$ after an hour whereas ship $A$ continues its sail, find the shortest distance between ship $A$ and ship $B$.</div></li><li><div class="list-item-paragraph">If ships $A$ and $B$, after an hour, then travel towards each other with their speeds unchanged, when will they meet?</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/1767430891990-jlobqui8iy.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MzA4OTE5OTAtamxvYnF1aThpeS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.hBnIrERIes_NsI4XHgKToKF2wjs-Ye4ip2aLRSnR7f0" style="max-width: min(220px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

Problem involving Bearing

<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>$A$, $B$ and $C$ are three points on level ground.<br>The bearing of $B$ from $A$ is $065{}^\circ$ and $\angle ABC=30{}^\circ$.<br>It is given that&nbsp;$C$ is $4$ km due south of $A$.<br>Calculate the</p><ol class="i-list"><li><div class="list-item-paragraph">bearing of $C$ from $B$,</div></li><li><div class="list-item-paragraph">distance of $BC$,</div></li><li><div class="list-item-paragraph">shortest distance between $A$ and $BC$.</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/1767430607419-uu5uyzezxb.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MzA2MDc0MTktdXU1dXl6ZXp4Yi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.mCiV1-k2BrgF5hEzC-OK-nkuXoSnYhtvojMZkY5u2Rg" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

<ol class="i-list"><li><div class="list-item-paragraph">Find the bearing of $X$ from $Y$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767430205035-x5x75ss6eb.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MzAyMDUwMzUteDV4NzVzczZlYi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.bvFBwLHfF3cRhOHit2PDy7NCGzaZArVVZ4G1EPp6Dm0" alt="List item image" class="aligncenter custom-width" style="max-width: min(150px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">Find the bearing of $X$ from $Y$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767430239594-hoaj3946ix.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MzAyMzk1OTQtaG9hajM5NDZpeC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.RQjZ0x7VWIwxEAxLFWmpj-DsIFNSDua2KfSp5XR_73A" alt="List item image" class="aligncenter custom-width" style="max-width: min(150px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">Find the bearing of $D$ from $A$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767430250576-tk8d22wdh7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MzAyNTA1NzYtdGs4ZDIyd2RoNy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.-uIsKR2SxPkkuLNUtbXwtPdYGCGhIJQnMCBNbNE-TwI" alt="List item image" class="aligncenter custom-width" style="max-width: min(150px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">Find the bearing of $E$ from $A$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767430262267-j4c3v7xfgpf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MzAyNjIyNjctajRjM3Y3eGZncGYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDczMywiZXhwIjoxNzkzNTA1MDMzfQ.IPf2OHb-7LsKHjdL7wdE23ZFMOc-4aNOwYpdB_TxDIQ" alt="List item image" class="aligncenter custom-width" style="max-width: min(150px, 100%); height: auto;" /></div></li></ol>

Finding the Bearing

<ol class="i-list"><li><div class="list-item-paragraph">Find the bearing of $B$ from $A$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767429048460-kmopf5pt8fg.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MjkwNDg0NjAta21vcGY1cHQ4ZmcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDczMywiZXhwIjoxNzkzNTA1MDMzfQ.r2iyy3HSzmXaYRNFFv2h1SxR4GcQroQ6XwFxz8UCirc" alt="List item image" class="aligncenter custom-width" style="max-width: min(140px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">Find the bearing of $C$ from $A$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767429062922-2r8qm9l5ziu.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MjkwNjI5MjItMnI4cW05bDV6aXUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDczMywiZXhwIjoxNzkzNTA1MDMzfQ.qBHIs57IRVPgWOOMWEasodp7ldm7Bo_okhBJWGGin4A" alt="List item image" class="aligncenter custom-width" style="max-width: min(140px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">Find the bearing of $D$ from $A$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767429087964-5jcpuggqte.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MjkwODc5NjQtNWpjcHVnZ3F0ZS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.r8X0W5nLpqwS6NR2EqMxMS9VkKZzWwWZTIQ-O5L-NZ8" alt="List item image" class="aligncenter custom-width" style="max-width: min(140px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">Find the bearing of $E$ from $A$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767429110619-oemj4ofeua.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MjkxMTA2MTktb2VtajRvZmV1YS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzMzLCJleHAiOjE3OTM1MDUwMzN9.NV4CdZfpa5nuU4N4_hd9WiPgYJb9tqiGZBOY0AvTd3A" alt="List item image" class="aligncenter custom-width" style="max-width: min(140px, 100%); height: auto;" /></div></li></ol>

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