A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>A cylindrical solid has a hemispherical top of the same radius $r\text{ cm}$. The cylinder has height $2r\text{ cm}$, and the volume of the entire solid is $576\pi\text{ cm}^3$.</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/1789189430874-y1vjytfj43.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODkxODk0MzA4NzQteTF2anl0Zmo0My5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzI1NTYzLCJleHAiOjE3OTI3MjU4NjN9.w7CUIJcN33BoJEFcnnxolmcp-lezD_RrrGo2qEbdbWE" style="max-width: min(150px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="sec4-em-2025-mgs-s-p2-q7-b3-i1" data-list-item-path="sec4-em-2025-mgs-s-p2-q7-b3-i1"><div class="list-item-paragraph">Show that&nbsp;$r=6$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="sec4-em-2025-mgs-s-p2-q7-b3-i2" data-list-item-path="sec4-em-2025-mgs-s-p2-q7-b3-i2"><div class="list-item-paragraph">The entire solid is melted to form a solid cone whose radius and height are equal. Find the total surface area of the cone.<span class="inline-marks">[4]</span></div></li></ol>

A cylindrical solid has a hemispherical top of the same radius $r\text{ cm}$. The cylinder has height $2r\text{ cm}$, and the volume of the entire solid is $576\pi\text{ cm}^3$.

The entire solid is melted to form a solid cone whose radius and height are equal. Find the total surface area of the cone.

Volume and Surface Area of Cylinders

Applications of Pythagoras’ Theorem

Volume and Surface Area of Spheres

Volume and Surface Area of Cones

Volume and Surface Area of Composite Solids

2025 MGS(S) PRELIMS P2 Q7

Volume and Surface Area of Prisms

Mensuration (Pyramid, Cones and Spheres)

Pythagoras’ Theorem

$UVWX$ is a square. Point $M$ is the midpoint of $XW$, and $N$ lies on $WV$ with $WN:NV=1:3$. Prove that $UMN$ is right-angled.

The Converse of Pythagoras Theorem

Right-angle proof from points on a square

<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/1781422877742-krek3774bzi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODE0MjI4Nzc3NDIta3JlazM3NzRiemkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.2a7iuOhhOFiaHIpf103zpKjtRTW9HI1ipfbP8AWUvJ4" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The diagram shows a circle, centre $O$. Points $A$, $B$, $D$ and $E$ lie on the circle. <br>The line $AD$ is a diameter of the circle and $DCP$ is the tangent of the circle at $D$. <br>The lines $EB$ and $AD$ intersect at point $Q$. <br>The chord $AB$ is produced to meet the tangent $DCP$ at $P$. <br>It is given that&nbsp;$\text{angle}\hspace{0.5em} CBP = \text{angle}\hspace{0.5em} DEB$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Show that&nbsp;$\text{angle}\hspace{0.5em} BCD = 90^\circ$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Show that&nbsp;$DP^2 = BP^2 + DA \times BC$.<span class="inline-marks">[4]</span></div></li></ol>

The diagram shows a circle, centre $O$. Points $A$, $B$, $D$ and $E$ lie on the circle. 
The line $AD$ is a diameter of the circle and $DCP$ is the tangent of the circle at $D$. 
The lines $EB$ and $AD$ intersect at point $Q$. 
The chord $AB$ is produced to meet the tangent $DCP$ at $P$. 
It is given that $\text{angle } CBP = \text{angle } DEB$.

Alternate Segment Theorem

Angles in the Same Segment

Similarity

Angles in a Semicircle and the Converse

2024 CGSS P1 Q13

Proofs in Plane Geometry

<p>In the diagram, $P$ is a point on the inside of the square $ABCD$ such that&nbsp;$PA = 2$ cm, $PB = 5$ cm and $PD = \sqrt{17}$ cm. Find the exact area of square $ABCD$.</p>
<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/1770789331534-kjh2q0njnp.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3ODkzMzE1MzQta2poMnEwbmpucC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzI1NTYzLCJleHAiOjE3OTI3MjU4NjN9.PJL8Gw-k4fM6CoTcJVv4tiZmEtqiCLAYmikwUwZcHhc" style="max-width: min(150px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

In the diagram, $P$ is a point on the inside of the square $ABCD$ such that $PA = 2$ cm, $PB = 5$ cm and $PD = \sqrt{17}$ cm. Find the exact area of square $ABCD$.

Geometric Transformations and Congruence

Perimeter and Area of Rectangles and Squares

Congruence

IP2U7 HOT Q3

Congruence and Similarity

Perimeter and Area of Plane Figures

<p>Given that the radius of the semicircle is $6$ cm, $ABCD$ and $CPQR$ are squares such that&nbsp;$A$, $Q$ are on the circumference, $B$, $C$, $P$ are on the diameter. Find the total area of the squares $ABCD$ and $CPQR$.</p>
<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/1770728173829-4yattd8463o.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3MjgxNzM4MjktNHlhdHRkODQ2M28ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.xrNOR0xuWpXg2NSVpIc1eRkwwH_YBp_Ds9923udQBH0" style="max-width: min(220px, 100%); height: auto; display: block;" alt="Two squares ABCD and CPQR stand on adjacent parts BC and CP of a diameter, with A and Q on the semicircle." class="alignright custom-width" /></figure><div class="image-caption" style="text-align: right;">Two squares ABCD and CPQR stand on adjacent parts BC and CP of a diameter, with A and Q on the semicircle.</div></div>

Given that the radius of the semicircle is $6$ cm, $ABCD$ and $CPQR$ are squares such that $A$, $Q$ are on the circumference, $B$, $C$, $P$ are on the diameter. Find the total area of the squares $ABCD$ and $CPQR$.

IP2U14 HOT Q4

Properties of Circle

<p>The distance between a pair of parallel lines, $a$ and $b$, is $4$ cm. The distance from point $A$ to line $a$ is $2$ cm, The distance from point $B$ to line $b$ is $3$ cm, and $AB=2\sqrt{30}$.</p>
<p>Find a point $M$ on line $a$ and a point $N$ on line $b$ such that&nbsp;$MN \perp a$ and the length of $AM + MN + NB$ is the shortest. Hence find the shortest length of $AM + NB$</p>
<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/1769504623407-vx6tpgp144e.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MDQ2MjM0MDctdng2dHBncDE0NGUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.I5NjDTYIEU-zSUOFXJUYkm2qRONHy8xVsBgcPTOUSvQ" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>
<p>As shown, $A$ lies outside the strip beyond line $a$, and $B$ lies outside the strip beyond line $b$.</p>

The distance between a pair of parallel lines, $a$ and $b$, is $4$ cm. The distance from point $A$ to line $a$ is $2$ cm, The distance from point $B$ to line $b$ is $3$ cm, and $AB=2\sqrt{30}$.

Find a point $M$ on line $a$ and a point $N$ on line $b$ such that $MN \perp a$ and the length of $AM + MN + NB$ is the shortest. Hence find the shortest length of $AM + NB$

As shown, $A$ lies outside the strip beyond line $a$, and $B$ lies outside the strip beyond line $b$.

IP1U13 SWII (Pythagoras) Q24

<p>In triangle $ABC$, $\angle B=90{}^\circ $, $AB=BC=6$ cm. $\Delta AEF$ is folded along $EF$, such that&nbsp;$A$ collides with $D$ and $BD:DC=1:2$. Find $EC$.</p>
<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/1769503806061-pwdjevzpvqs.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MDM4MDYwNjEtcHdkamV2enB2cXMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.yyJuomUNlYl9s7vnc_7ZIgh7Zmzxvcrsz4Xf_gkYv_A" style="max-width: min(150px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

In triangle $ABC$, $\angle B=90{}^\circ $, $AB=BC=6$ cm. $\Delta AEF$ is folded along $EF$, such that $A$ collides with $D$ and $BD:DC=1:2$. Find $EC$.

IP1U13 SWII (Pythagoras) Q22

<p>$AD$ is a median of triangle $ABC$, $\angle ADC=45{}^\circ$, $\Delta ADC$ is folded along $AD$ such that&nbsp;$C$ is mapped onto $C'$. Given that&nbsp;$BC=2$ cm, find $BC'$.</p>
<p>Note: The median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side.</p>
<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/1769503418493-w422o75v20p.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MDM0MTg0OTMtdzQyMm83NXYyMHAucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.INZpgDDFQwp0otv6ohbEY07_4LIn6Va-WMiPxPEgH3A" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

$AD$ is a median of triangle $ABC$, $\angle ADC=45{}^\circ$, $\Delta ADC$ is folded along $AD$ such that $C$ is mapped onto $C'$. Given that $BC=2$ cm, find $BC'$.

Note: The median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side.

IP1U13 SWII (Pythagoras) Q21

<p>In square $ABCD$, $E$ is a point on $BC$. $\Delta DCE$ is folded along $DE$ onto $DB$, such that&nbsp;$C$ collides with $F$. Find $EB:CE$.</p>
<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/1769502110590-m1coqsyyhin.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MDIxMTA1OTAtbTFjb3FzeXloaW4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.Mp60myoqNhgnWRgOKMFXjs5bYs95B_8zVgiSAS0LQM8" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

In square $ABCD$, $E$ is a point on $BC$. $\Delta DCE$ is folded along $DE$ onto $DB$, such that $C$ collides with $F$. Find $EB:CE$.

IP1U13 SWII (Pythagoras) Q20

<p>In right-angled triangle $ABC$, $AC=6$ cm, $BC=8$ cm. $\Delta ACD$ is folded along $AD$ onto $AB$ such that&nbsp;$C$ collides with $E$. Find the length of $CD$.</p>
<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/1769501375287-owwcm6vmk9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MDEzNzUyODctb3d3Y202dm1rOS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzI1NTYzLCJleHAiOjE3OTI3MjU4NjN9.pOmz3evqVX8_l11UQf_4q7KqG56YdA742IRy2kQcYWE" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

In right-angled triangle $ABC$, $AC=6$ cm, $BC=8$ cm. $\Delta ACD$ is folded along $AD$ onto $AB$ such that $C$ collides with $E$. Find the length of $CD$.

IP1U13 SWII (Pythagoras) Q19

<p>A rectangle with dimensions $4$ cm and $8$ cm is folded along $EF$, such that&nbsp;$C$ collides with $A$. Find the length of $EB$.</p>
<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/1769500106202-hlcu4sh5fo9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MDAxMDYyMDItaGxjdTRzaDVmbzkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.kUqeKvSIzY3GTi40kEgdZ3D38m0vZ9vupIcqOt9b16o" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

A rectangle with dimensions $4$ cm and $8$ cm is folded along $EF$, such that $C$ collides with $A$. Find the length of $EB$.

IP1U13 SWII (Pythagoras) Q18

<p>In the diagram, all quadrilaterals are squares, all triangles are right-angled triangles. The side of the largest square is $7$ cm. Find the total area of the squares $A$, $B$, $C$ and $D$.</p>
<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/1768920874096-f0adeqgeidq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MjA4NzQwOTYtZjBhZGVxZ2VpZHEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.bdwBxgTOCIimxsA3VB0itbi4qwhK2Zjx-I5b6feilLI" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

In the diagram, all quadrilaterals are squares, all triangles are right-angled triangles. The side of the largest square is $7$ cm. Find the total area of the squares $A$, $B$, $C$ and $D$.

IP1U13 SWII (Pythagoras) Q17

<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 triangle $ABC$, $\angle ACB=90^\circ$.<br>Draw semicircles with $AB$, $AC$ and $BC$ as diameters, respectively.<br>Find the ratio of the area of the shaded region and the area of triangle $ABC$.<br></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/sign/question-images/pasted/1768919862659-3m31roun6p2.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTk4NjI2NTktM20zMXJvdW42cDIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.hvAfXtRuSSisML8gt91ugnYaKeN5AbmFmwYS5DQtai0" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

Perimeter and Area of Composite Figures

IP1U13 SWII (Pythagoras) Q15

<p>Triangle $ABC$ is a right-angled triangle with $AB=BC=10$ cm.<br>$OBDC$ is a semicircle with centre $O$ and $D$ is the midpoint of the arc $BDC$.<br>Calculate the area and perimeter of the shaded region, to $3$ significant figures.</p>
<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/1768918607800-zywmzs3ar6.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTg2MDc4MDAtenl3bXpzM2FyNi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzI1NTYzLCJleHAiOjE3OTI3MjU4NjN9.Ny4Dd3--XpfZqvw5kwDean55PTDcmY7GReLmQT622cM" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

Triangle $ABC$ is a right-angled triangle with $AB=BC=10$ cm.
$OBDC$ is a semicircle with centre $O$ and $D$ is the midpoint of the arc $BDC$.
Calculate the area and perimeter of the shaded region, to $3$ significant figures.

Area of Sector

Arc Length

IP1U13 SWII (Pythagoras) Q13

Arc Length and Area of Sector

<p>In the right-angled triangle shown, $\angle ACB=90^\circ$, $AC=15\text{ cm}$ and $BC=8\text{ cm}$. The printed $\angle BAC=28^\circ$ is rounded to the nearest degree. Use the exact side lengths and unrounded calculated angles for the answers to two decimal places.</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 style="text-align: left">Arc $MC$ has centre $A$ and arc $NC$ has centre $B$.</p><p style="text-align: left">Find, correct to $2$ decimal places:</p><ol class="abc-list"><li data-list-item-id="item-1768917404334-t5neuvba9" data-list-item-path="item-1768917404334-t5neuvba9"><div class="list-item-paragraph">the perimeter of the shaded region,</div></li><li data-list-item-id="item-1768917404934-lbywhm5fi" data-list-item-path="item-1768917404934-lbywhm5fi"><div class="list-item-paragraph">the area of the shaded region.</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/1768917502087-nynvjmzed7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTc1MDIwODctbnludmptemVkNy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzI1NTYzLCJleHAiOjE3OTI3MjU4NjN9.xIwkep6_z_uwM1VSI-kD-BPXl7D2f-8i2da0-RiszEw" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the right-angled triangle shown, $\angle ACB=90^\circ$, $AC=15\text{ cm}$ and $BC=8\text{ cm}$. The printed $\angle BAC=28^\circ$ is rounded to the nearest degree. Use the exact side lengths and unrounded calculated angles for the answers to two decimal places.

Basics of Trigonometric Ratios

IP1U13 SWII (Pythagoras) Q12

Trigonometry

<p>$8$ circles each of diameter $6\text{ cm}$ are cut off from a rectangular piece of paper $ABCD$. The points $E$ to $P$ are on the perimeter of the paper and the circumference of the circles.</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">Write down the length and breadth of rectangle $ABCD$.</div></li><li><div class="list-item-paragraph">Calculate, to $3$ significant figures, the length of</div><ol class="i-list"><li><div class="list-item-paragraph">$EL$,</div></li><li><div class="list-item-paragraph">$NP$,</div></li><li><div class="list-item-paragraph">$AJ$,</div></li><li><div class="list-item-paragraph">$BN$.</div></li></ol></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-end;"><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/1768916151939-g9v7yge4b3j.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTYxNTE5MzktZzl2N3lnZTRiM2oucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.B7VBcmnnIaXFkxrlOgU6n8wsApqCKfhM_id2g71lv2Y" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

$8$ circles each of diameter $6\text{ cm}$ are cut off from a rectangular piece of paper $ABCD$. The points $E$ to $P$ are on the perimeter of the paper and the circumference of the circles.

IP1U13 SWII (Pythagoras) Q11

<p>$WXYZ$ is a rectangle in which $WX=12\text{ cm}$ and $WZ=8\text{ cm}$.<br>$T$ is a point on $WX$ such that $YZT$ is an isosceles triangle whose equal sides are $YZ$ and $ZT$.<br>Find $YT$ correct to $3$ significant figures.<br></p>
<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/1768915153139-bp5nqhyog74.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTUxNTMxMzktYnA1bnFoeW9nNzQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.9YxyNwxRbRaWsGqWDMi9xn5i--XYbPzrfXXJRg3iZmM" style="max-width: min(170px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

$WXYZ$ is a rectangle in which $WX=12\text{ cm}$ and $WZ=8\text{ cm}$.
$T$ is a point on $WX$ such that $YZT$ is an isosceles triangle whose equal sides are $YZ$ and $ZT$.
Find $YT$ correct to $3$ significant figures.


IP1U13 SWII (Pythagoras) Q10

<p>Find the length of $AE$, in terms of $x$, given that $AB=BC=CD=DE=x\text{ cm}$.</p>
<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/1768914432533-1moww32ejgf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTQ0MzI1MzMtMW1vd3czMmVqZ2YucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.kqfkwTO6DtaoHerTr3O_yVyWYZpJ1dZG7Wau-oHLv2k" style="max-width: min(150px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

Find the length of $AE$, in terms of $x$, given that $AB=BC=CD=DE=x\text{ cm}$.

IP1U13 SWII (Pythagoras) Q9

<ol class="abc-list"><li><div class="list-item-paragraph">Show that a triangle is right-angled if the sides are $a={{m}^{2}}-{{n}^{2}}$, $b=2mn$ and $c={{m}^{2}}+{{n}^{2}}$ where $m$ and $n$ are 	positive,	and $m&gt;n$.</div></li><li><div class="list-item-paragraph">In triangle $ABC$, $AB=AC$,$P$ is an arbitrary point on $BC$. Show that $A{{B}^{2}}-A{{P}^{2}}=PB\times PC$.</div></li></ol>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768913649318-3k13v5kla8l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTM2NDkzMTgtM2sxM3Y1a2xhOGwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTU2MywiZXhwIjoxNzkyNzI1ODYzfQ.ZwcToLmathwjA-FY6BLOaCEXf_qrF4Hp02Ht7ie25ZA" style="max-width: min(150px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

Algebraic Proof Using Identities

IP1U13 SWII (Pythagoras) Q8

Algebraic Expressions, Identities and Formulae

<p>A $25\text{ m}$ ladder leans against a vertical wall.<br>The foot of the ladder is $20\text{ m}$ from the base of the wall.<br>If the foot is moved $13\text{ m}$ closer to the wall, how far does the top of the ladder move up the wall?<br></p>

A $25\text{ m}$ ladder leans against a vertical wall.
The foot of the ladder is $20\text{ m}$ from the base of the wall.
If the foot is moved $13\text{ m}$ closer to the wall, how far does the top of the ladder move up the wall?


Math Question Bank