CAIE IGCSE Mathematics Extended E4.4 Similarity

Covered syllabus points

E4.4.1 Calculate lengths of similar shapes.E4.4.2 Use the relationships between lengths and areas of similar shapes and E4.4.3 Solve problems and give simple explanations involving similarity

[Maximum number: 2]

NOT TO SCALE

The diagram shows a cyclic quadrilateral ABCD. BD and AC intersect at X.

In the diagram, triangle ADX is similar to triangle BCX.
$B C=4.5 \mathrm{~cm}, A D=9 \mathrm{~cm}$ and $C X=3.3 \mathrm{~cm}$ .

Work out XD.
X D= cm [2]

6.6

M1 for $\frac{4.5}{9}=\frac{3.3}{XD}$ oe.

[Maximum number: 3]

Figure for Question 2 — CAIE IGCSE Mathematics Extended

NOT TO
SCALE

The two shapes are mathematically similar.
The area of the larger shape is $36 \mathrm{~cm}^{2}$ and the area of the smaller shape is $25 \mathrm{~cm}^{2}$ .
The height of the larger shape is 9 cm and the height of the smaller shape is $x \mathrm{~cm}$ .

Find the value of x.

$\frac{15}{2}$ oe

M2 for $\frac9x=\frac65$ oe, or $x^2=\frac{25}{36}\times9^2$ oe or better.
M1 for $\sqrt{\frac{36}{25}}$ or $\sqrt{\frac{25}{36}}$ oe or better, or for $\left(\frac9x\right)^2=\frac{36}{25}$ or $\left(\frac{x}{9}\right)^2=\frac{25}{36}$ oe or better.

[Maximum number: 4]

NOT TO SCALE

G, K and M lie on a circle, centre O.
FGH is a tangent to the circle at G and MG is parallel to OH.
Show that triangle GKM is mathematically similar to triangle OHG.
Give a geometrical reason for each statement you make.

Two pairs of equal angles identified with fully correct reasons
M3
M2 for one pair of equal angles identified with fully correct reasons
Examples:
$KMG=90^\circ$ (angle in semicircle) and $OGH=90^\circ$ (angle between tangent and radius)
OR KMG=OGH (alternate segment)
OR GOH=MGK (alternate angles)
OR angle FGM= angle GHO (corresponding), angle FGM=GKM (alternate segment), and angle H= angle K
M1 for $KMG=90^\circ$ angle in semicircle or $OGH=90^\circ$ angle between tangent and radius
A1 dep on M3 with no incorrect work: two or three pairs of angles equal, so triangles are similar

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CAIE IGCSE Mathematics Extended E4.4 Similarity Topic Practice

Covered syllabus points

Question 1

Question 2

Question 3