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<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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MDQ2MjM0MDctdng2dHBncDE0NGUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ.UwQ9-J-2f_lf96laSgV3gqfwSVdLMPkcP_n6GTfOLbA" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

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 $AM + NB$

Applications of Pythagoras’ Theorem

IP1U13 SWII (Pythagoras) Q24

Pythagoras’ Theorem

<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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MDM4MDYwNjEtcHdkamV2enB2cXMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ._Kka20RGPtdvUft-GaLxvhNHSgU3Zx0zHvaOfFvUIWI" 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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MDM0MTg0OTMtdzQyMm83NXYyMHAucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ.oGx6uYNr3o93etUcbtktzx2VZvm_cCipTfpqwnL1gs0" 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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MDIxMTA1OTAtbTFjb3FzeXloaW4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ.UikUyNgNxNIe-G_CMtMPjkAUUD33DtTc_ORk34YmGD4" 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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MDEzNzUyODctb3d3Y202dm1rOS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzIzOTQ5LCJleHAiOjE3ODU3MjQyNDl9.1HLuuNyPqXkhpB4IgCxhAS5LwYsMfCKH3XDk6qB0oqE" 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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MDAxMDYyMDItaGxjdTRzaDVmbzkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ.cdgEv1ozbqtNvAF1FvD0ka0az-DDUPH2ViIRMnlK1pU" 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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MjA4NzQwOTYtZjBhZGVxZ2VpZHEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ.-VXLf6zZfyWe86hmVPIiZ1KRb75M3z152UX6jIPa0k4" 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

<p>A cube sits tightly inside a sphere such that all $8$ corners touch the sphere.<br>Find the side length of the cube if the cross-sectional area of the sphere was $\pi \text{ c}{{\text{m}}^{2}}$.</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/1768920383306-39p3v12cqq9.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MjAzODMzMDYtMzlwM3YxMmNxcTkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ.JaKHYT-SLk4pSDC_8Toi1MYH6xBzkiEJnZFOj3pXBNw" style="max-width: min(150px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

A cube sits tightly inside a sphere such that all $8$ corners touch the sphere.
Find the side length of the cube if the cross-sectional area of the sphere was $\pi \text{ c}{{\text{m}}^{2}}$.

IP1U13 SWII (Pythagoras) Q16

<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 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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTk4NjI2NTktM20zMXJvdW42cDIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ.p3c8zz9zt_iLquL6T8Yp6mpelSaXuTjjCNRV9LCmb5s" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

IP1U13 SWII (Pythagoras) Q15

<p>A cylinder is three times as high as it is wide.</p>
<p>The length of the cylinder’s diagonal is $20\text{ cm}$.</p>
<p>Find the <strong>EXACT</strong> surface area of the top of the cylinder (as shaded in the image).</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>(Note: Your solution will have the symbol $\pi $ in it.)</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/1768919168947-al2ixqh65g.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTkxNjg5NDctYWwyaXhxaDY1Zy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzIzOTQ5LCJleHAiOjE3ODU3MjQyNDl9._-LUuUZg1JcXWOcMuPmzXrewo6UgBuL-XS8WbCfPCu8" style="max-width: min(160px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

A cylinder is three times as high as it is wide.

The length of the cylinder’s diagonal is $20\text{ cm}$.

Find the **EXACT** surface area of the top of the cylinder (as shaded in the image).

IP1U13 SWII (Pythagoras) Q14

<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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTg2MDc4MDAtenl3bXpzM2FyNi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzIzOTQ5LCJleHAiOjE3ODU3MjQyNDl9.nKSUD8Vu2-XHz7rJdkG1t4pPGio5bAQHKHTWzPSJrIg" 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.

IP1U13 SWII (Pythagoras) Q13

<p>In the right-angled triangle shown below, $AC=15$cm, $BC=8$cm, $\angle BAC={{28}^{\circ }}$.</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;"><p>Arc $MC$ has centre $A$ and arc $NC$ has centre $B$.</p><p>Find, correct to $2$ decimal places:</p><ol class="abc-list"><li><div class="list-item-paragraph">the perimeter of the shaded region,</div></li><li><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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTc1MDIwODctbnludmptemVkNy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzIzOTQ5LCJleHAiOjE3ODU3MjQyNDl9.gMYaifBOTTPa6iNypNjpIG50cxNtSF3iRJqzE_2MUiI" 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 below, $AC=15$cm, $BC=8$cm, $\angle BAC={{28}^{\circ }}$.

IP1U13 SWII (Pythagoras) Q12

<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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTYxNTE5MzktZzl2N3lnZTRiM2oucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ.ZCEfDlIvYm43Z84fdEQAzsuDwRUa8C8zuM8-RZAojyc" 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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTUxNTMxMzktYnA1bnFoeW9nNzQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ.tffZAcKvtpZ2wI2lgKq5CFGZsMmKMHWg6wP78fyWdhc" 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.


Basics of Pythagoras’ Theorem

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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTQ0MzI1MzMtMW1vd3czMmVqZ2YucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ.XO9UWWmxOy_5knUHSx6NljO3wIS3_ThioGL8pUZ3Nns" 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=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTM2NDkzMTgtM2sxM3Y1a2xhOGwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ.ftzEgvhFRaoJ-aAwvxEaYQ1V6m4y2EvllZxyVQ4k57g" style="max-width: min(150px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

The Converse of Pythagoras Theorem

IP1U13 SWII (Pythagoras) Q8

<p>A pencil box, in the shape of a rectangular prism, measures $16\text{ cm}\times 12\text{ cm}\times 8\text{ cm}$.<br></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>Find the length of the longest pencil that would fit inside the box.<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/1768912850367-tljf1lbpp1.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTI4NTAzNjctdGxqZjFsYnBwMS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzIzOTQ5LCJleHAiOjE3ODU3MjQyNDl9.H4D7RWCjaY3JWIG8_9VRXS7Yq9EWidJmkvgXDJdzdqY" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

A pencil box, in the shape of a rectangular prism, measures $16\text{ cm}\times 12\text{ cm}\times 8\text{ cm}$.


IP1U13 SWII (Pythagoras) Q7

<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?


IP1U13 SWII (Pythagoras) Q6

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the area of an equilateral triangle with $2$ cm sides.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the area of an equilateral triangle with $10$ cm sides.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the area of a regular hexagon which has $4$ cm sides.</span></div></li></ol>

Find the area of an equilateral triangle with $2$ cm sides.

Find the area of an equilateral triangle with $10$ cm sides.

Find the area of a regular hexagon which has $4$ cm sides.

IP1U13 SWII (Pythagoras) Q5

<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>Below are two triangles, $ABC$ and $BCD$.<br>Find the length of $BD$.<br>In triangle $ABC$, $AB=7$cm, $AC=6$cm, $\angle CAB={{90}^{\circ }}$. <br>In triangle $BCD$, $CD=11$cm, $\angle DCB={{90}^{\circ }}$.<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/1768910307639-izahjitqk1r.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTAzMDc2MzktaXphaGppdHFrMXIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcyMzk0OSwiZXhwIjoxNzg1NzI0MjQ5fQ.4K3UuDONifUIWfdHtyjNnZVNgwEDfQEidF-D8XHAcH0" style="max-width: min(120px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

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