MATHEMATICS O LEVEL(FORM TWO) NOTES – SIMILARITY AND ENLARGEMENT

SIMILARITY AND ENLARGEMENT

Similar figures:

Two polygons are said to be similar if they have the same shape but not necessarily the same size.

When two figures are similar to each other, the corresponding angles are equal and the ratios of corresponding sides are equal.

SIMILAR TRIANGLE

Triangles are similar when their corresponding angles are equal or corresponding sides proportional. Consider the figure below:

Since corresponding angles are equal, the two triangles are similar.

Also:

Since the ratio of corresponding sides are equal, the two triangles are similar.

Note

() is a sign of similarity. From above, ABC PQR.

Examples

1. Given that SLK NFR, identify all the corresponding angles and corresponding sides.

Corresponding sides:

3. One rectangle has length 10cm and width 5cm. The second rectangle has length 12cm and width 4cm. Are the two rectangles similar? Explain.

Solution:

Therefore; the two rectangles are not similar because the ratio of corresponding sides are not proportional.

4. A rectangle has length 16cm and width 23cm. A second rectangle has length 12cm and width 9cm. Are the two rectangles similar? Explain.

Solution:

Therefore; The rectangles are not similar because the ratio of corresponding sides are not proportional.

Conditions for two triangles to be similar:

1. Corresponding angles are equal or corresponding sides proportional.

For other polygons:

• Corresponding angles equal and corresponding sides proportional.

QUESTIONS:

a) Given that PQR LMN and that PQR ABC identify the corresponding angles and sides between ABC and LMN.

Solution

a) Name the triangles which are similar.

b) Identify the corresponding angles.

Solution:

The triangles ABT and KLS are similar.

8. Name the triangles which are similar to ADC.

10. Which of the following figures are always similar?

• a) circles
• b) Hexagons
• c) squares
• d) Rhombuses
• e) Rectangles
• f) Congruent polygons

Solution:

The figures which are always similar:

• a) circles
• b) squares

Exercise 1

M < AEF = 42°

M < AFE = ?

90° – 42° = 48°

M < AFE = 48°

INTERCEPT THEOREM

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

AAA – Similarity theorem

If a correspondence between two triangles is such that two pairs of corresponding angles are equal, then the two triangles are similar.

SSS – similarity Theorem

If the two triangles are such that corresponding sides are proportional, then the triangles are similar.

SAS – Similarities theorem

If the two triangles are such that two pairs of corresponding sides are proportional and the included angles are congruent, then the triangles are similar.

PROPERTIES OF SIMILAR TRIANGLES

From the previous discussion, properties of similar triangles can be summarized as:

1. Corresponding angles of similar triangles are equal.
2. Corresponding sides of similar triangles are proportional.
3. Two triangles are similar if two angles of one triangle are respectively equal to two corresponding angles of the other.
4. Two triangles are similar if an angle of one triangle equals an angle of the other and the sides including these angles are proportional.

ENLARGEMENT

Scale enlargement

Scale is a ratio between measurements of a drawing to the actual measurement.

It is normally stated in the form 1:…, for example, if a scale of a map is 1:20000, then 1 unit on the map represents 20000 units on the ground.

Scale =

Examples of scales

1. Find the length of the drawing that represents:

• a) 1 stem when the scale is 1:500,000

Solution:

1:500,000 means 1 cm on the drawing represents 500,000 cm on the actual distance.

500,000x = 1,500,000

x = 3 cm

The drawing length is 3 cm.

• b) 45 km when scale is 1 cm to 900 m

Solution:

Scale = 1:90,000

x = 50 cm

The drawing distance is 50 cm.

• a) 3.5 cm when the scale is 1:5000 m

Solution:

y = 5000 × 3.5

y = 17,500 cm = 175 m

The distance is 175 m.

• b) 1.8 mm when the scale is 1 cm to 500 metres

Solution:

v = 0.18 × 50,000

v = 9,000 cm = 90 m

The actual length is 90 m.

Exercise:

1. Find the length of the drawing that represents:

• a) 200 m when the scale is 1 cm to 50 meters

Scale = 1:50

x = 4 cm

The length of drawing = 4 cm.

• b) 1.5 m when the scale is 1 cm to 100 meters

x = 15 cm

The length of drawing = 15 cm.

• d) 1600 km when the scale is 1 mm to 1 km

x = 1.6 mm

The length of drawing is 1.6 mm.

• e) 10 m when the scale is 1:500

x = 2 cm

The length of drawing = 2 cm.

• a) 13.15 mm when the scale is 1:4000

x = 0.0032875 mm

• b) 3.78 cm when the scale is 1 mm to 50 km

x = 11.5 cm

The corresponding width of drawing = 11.5 cm.

Solution:

Scale =

x = 11.5 cm

The corresponding width of drawing = 11.5 cm.

ENLARGEMENT

When two figures are similar, one can be considered the enlargement of the other.

(a)

(b) Square ABCD is the enlargement of PQRS.

(c) The larger circle is the enlargement of the smaller circle.

Example

1. State whether ABCD is the enlargement of PQRS.

Solution:

Since the corresponding sides are in the ratio of 2:1 and corresponding angles are equal, then ABCD is an enlargement of PQRS.

Scale factor:

If two polygons are similar and the ratio of their corresponding sides is 5:3, then the enlargement scale is 5/3.

Example

Find the scale of enlargement, hence calculate:

Solution:

Scale factor for areas

If two polygons have a scale factor of K, then the ratio of the areas is K².

Example

If ABS VST and the area of STV is 6 square cm, find the area of ABC.

Exercise

1. Two triangles are similar but not congruent. Is one the enlargement of the other? One triangle is the enlargement of the other.
2. The length of a rectangle is twice the length of another rectangle. Is one necessarily an enlargement of the other? Explain. No, since the widths are not necessarily in the same proportion as the lengths.
3. In the figure below, show that PQR is not an enlargement of DEF.

5. Triangle XYZ is similar to triangle ABC and XY = 8 cm. If the area of triangle XYZ is 24 cm² and the area of triangle ABC is 96 cm², calculate the length of AB.