Similarity & Geometrical Proof (Edexcel IGCSE Maths): Exam Questions

$PQRST$ is a regular pentagon. $R,U$ and $T$ are points on a circle, centre $O$. $QR$ and $PT$ are tangents to the circle. $RSU$ is a straight line.

Prove that $ST=UT$.

$A,B,C$ and $D$ are four points on the circumference of a circle.

$AEC$ and $BED$ are straight lines. Prove that triangle $ABE$ and triangle $DCE$ are similar. You must give reasons for each stage of your working.

$ABC$ and $AED$ are straight lines. $BE$ and $CD$ are parallel

. $BE=1.5\mathrm{cm}.\,CD=6\mathrm{cm}.\,AD=5\mathrm{cm}.$

Calculate the length of $ED$.

Quadrilaterals ABCD and LMNP are mathematically similar.

Angle A = angle *L * Angle B = angle *M * Angle C = angle N Angle D = angle P

Work out the length of LP.

Work out the length of BC.

The two triangles in the diagram are similar.

There are two possible values of $x$.

Work out each of these values. State any assumptions you make in your working.

Steve has a photo and a rectangular piece of card.

The photo is 16 cm by 10 cm. The card is 30 cm by 15 cm.

Steve cuts the card along the dotted line shown in the diagram below.

Steve throws away the piece of card that is 15 cm by $x$ cm. The piece of card he has left is mathematically similar to the photo.

Work out the value of $x$.

ABC and EDC are straight lines. EA is parallel to DB.

EC = 8.1 cm. DC = 5.4 cm. DB = 2.6 cm,

Work out the length of AE.

AC = 6.15 cm.

Work out the length of AB.

$ABC$ and $ABD$ are two right-angled triangles.

$\mathrm{Angle}BAC=\mathrm{angle}ADB=90^{\circ}\,\,AB=13\mathrm{cm}\,DB=5\mathrm{cm}$

Work out the length of $CB$.

The diagram shows the side view of a step ladder with a horizontal strut of length 48 cm.

The strut is one third of the way up the ladder.

The symmetrical cross section of the ladder shows two similar triangles.

Work out the vertical height, $h$ cm, of the ladder.

...........................cm

Rectangle $ABCD$ is mathematically similar to rectangle $DAEF$.

$AB=10\mathrm{cm}.\,AD=4\mathrm{cm}.$

Work out the area of rectangle $DAEF$.

The diagram shows two water towers in Kuwait.

The real height of tower $A$ is $187m$. The real height of tower $B$ is $147m$.

Ahmed makes a scale model of both towers.

The height of tower $A$ on the scale model is $90\mathrm{cm}$.

Work out the height of tower $B$ on the scale model. Give your answer correct to the nearest centimetre.

Triangle $ABC$ is similar to triangle $PQR$

$AB=4\mathrm{cm}$ $PQ=12\mathrm{cm}$ $RQ=16.5\mathrm{cm}$ $AC=x\mathrm{cm}$ $PR=y\mathrm{cm}$

Calculate the length of $BC$

Write down an expression for $y$ in terms of $x$

y = ......................................

$C$ lies on a circle with diameter $AD$. $B$ lies on $AC$ and $E$ lies on $AD$ such that $BE$ is parallel to $CD$. $AB$ = 21 cm, $CD$ = 18 cm and $BE$ = 13.5 cm.

Work out the radius of the circle.

............................................ cm

PQR and PTS are straight lines. Angle PTQ = Angle PSR = 90^o. QT = 4 cm RS = 12 cm TS = 10 cm

Work out the area of the trapezium QRST.

Work out the length of PT.

$APB$and $CPD$are chords of a circle.

$AP$ = 9 cm $PB$ = 6 cm $CP$ = 8 cm

Calculate the length of $PD$.

......................cm

$ABC$ and $DEF$ are similar triangles.

Work out the length of $AB$.

Given that $BC=54\mathrm{cm}$, work out the length of $EF$.

The diagonals of the cyclic quadrilateral ABCD intersect at X.

Explain why triangle ADX is similar to triangle BCX. Give a reason for each statement you make.

*AD *= 10 cm, *BC *= 8 cm, BX = 5 cm, *CX * = 7 cm.

Calculate DX.

DX = ........................................... cm

$PTQ$is a diameter of a circle. $RTS$ is a chord of the circle.

Calculate the radius of the circle.

........................................... cm

$PRT$ and $QRS$ are similar triangles.

Which of these is equivalent to $\frac{QR}{PR}$?

Circle your answer.

The diagram shows two cylinders, $A$ and $B$

Cylinder $A$ has height 1.6 m and radius 0.56 m.

Cylinder $B$ is mathematically similar to cylinder $A$. The height of cylinder $B$ is 0.6 m. Work out the radius of cylinder $B$.

....................................................... m

A solid metal cone has radius 10 cm and height 36 cm. Calculate the volume of this cone.

[The volume, $V$, of a cone with radius $r$ and height $h$ is $V=\frac{1}{3}πr^{2}h$.]

......................................... cm³

The cone is cut, parallel to its base, to give a smaller cone.

The volume of the smaller cone is half the volume of the original cone.

The smaller cone is melted down to make two different spheres.

The ratio of the radii of these two spheres is 1 : 2.

Calculate the radius of the smaller sphere.

[The volume, $V$, of a sphere with radius $r$ is $V=\frac{4}{3}πr^{3}$.]

.......................................... cm

The diagram shows a triangle and a trapezium.

Prove that $a=b$

$A,B$and $C$ are points on a circle.

• $BC$ bisects angle $ABQ$.
• $PBQ$ is a tangent to the circle.

Angle $CBQ=x$

Prove that $AC=BC$

$ABC$and $DEF$are similar triangles.

Work out the length of $DF$.

.......................$\mathrm{cm}$

Work out the length of $BC$.

....................$\mathrm{cm}$

The diagram below shows two right-angled triangles.

Prove that triangles PQS and QRS are similar.

The diagram shows triangle ABC. CD is parallel to AB. A, C and E lie in a straight line. Angles of size $a^{\circ},b^{\circ}$and $c^{\circ}$ are shown.

Insert $a^{\circ},b^{\circ}$° or $c^{\circ}$ to make this statement true. Give a reason for your answer.

Angle DCE = ......... because ....................................................................................................

Use the diagram and the answer to part (a) to show that the angles of a triangle add up to 180°.

Give a reason for each statement you make.

$ABC$and $DEF$are similar triangles.

Work out the length of $DE.$

........................$\mathrm{cm}$

In the diagram, $PQ$ is parallel to $BC$. $APB$ and $AQC$ are straight lines. $PQ$ = 8 cm, $BC$ = 10 cm and $AB$ = 9cm.

Calculate $PB$.

$PB$ = .......................................... cm

In the diagram AB is parallel to CD. AED and BEC are straight lines.

Prove that triangle ABE is similar to triangle CDE.

These two triangles are similar.

Work out the value of $a$.

............................cm

Anna estimates the height of a tree.

Anna holds a ruler vertically so the height of the tree is exactly covered by the ruler. She is 20 metres from the tree. The ruler is 30cm long. The horizontal distance from her eyes to the ruler is 60 cm.

Calculate an estimate of the height of the tree.

.......................... m

Give two reasons why this method may not be suitable to estimate the height of a very tall building.

Here are two right-angled triangles.

Circle the value of $y$.

In the diagram, AB and CD are parallel. AD and BC intersect at right angles at the point X. AB = 10 cm, CD = 5 cm, AX = 8 cm and BX = 6 cm.

Use similar triangles to calculate DX.

DX = ........................................... cm

Triangle $ABC$ is similar to triangle $PQR$.

Find $PQ$.

$PQ=$ ......................................... cm