CIE IGCSE Mathematics (0580) Similarity Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Similarity Study Notes
LEARNING OBJECTIVE
- Understanding Similar Figures
Key Concepts:
- Similarity: Length, Area, and Volume
- Similarity in Triangles
Similarity
Two shapes are similar if they have the same shape but not necessarily the same size. Their corresponding angles are equal, and their corresponding sides are in the same ratio.
Key Facts:
- All angles in similar shapes are equal.
- Corresponding sides are in the same ratio (called the scale factor).
- If the scale factor is \( k \), then:
- Lengths are multiplied by \( k \)
- Areas are multiplied by \( k^2 \)
- Volumes are multiplied by \( k^3 \)
Formula:
\( \text{Scale Factor} = \frac{\text{Length in enlarged shape}}{\text{Length in original shape}} \)
Example:
Triangle A and Triangle B are similar. Triangle A has side lengths 6 cm, 8 cm, and 10 cm. The shortest side of Triangle B is 9 cm. Find the lengths of the other two sides of Triangle B.
▶️ Answer/Explanation
Step 1: Find scale factor.
Shortest side of A = 6 cm, shortest side of B = 9 cm
Scale factor = \( \frac{9}{6} = 1.5 \)
Step 2: Multiply other sides by 1.5
Second side: \( 8 \times 1.5 = 12 \ \text{cm} \)
Third side: \( 10 \times 1.5 = 15 \ \text{cm} \)
Answer: The other two sides of Triangle B are 12 cm and 15 cm.
Example:
A model car is built to a scale of 1:20. If the length of the real car is 4.8 m, what is the length of the model car?
▶️ Answer/Explanation
Scale 1:20 means model : real = 1 : 20
Model length = \( \frac{4.8}{20} = 0.24 \ \text{m} = 24 \ \text{cm} \)
Answer: The model car is 24 cm long.
Similarity: Length, Area, and Volume
Similarity: Length, Area, and Volume
When two shapes are similar, their corresponding lengths, areas, and volumes are related through powers of the scale factor.
Key Relationships:
- Length ratio: If the scale factor between two similar shapes is \( k \), then all corresponding lengths are in the ratio \( k:1 \).
- Area ratio: The ratio of areas is \( k^2:1 \).
- Volume ratio: The ratio of volumes is \( k^3:1 \).
These relationships apply to:
- 2D shapes (length and area)
- 3D solids (length, surface area, and volume)
Example:
Two similar triangles have side lengths in the ratio 3:5. Find the ratio of their areas.
▶️ Answer/Explanation
Length ratio = \( 3:5 \)
Area ratio = \( 3^2:5^2 = 9:25 \)
Example:
Two similar spheres have radii in the ratio 2:3. What is the ratio of their volumes?
▶️ Answer/Explanation
Length ratio = \( 2:3 \)
Volume ratio = \( 2^3:3^3 = 8:27 \)
Example:
The surface area of a small cube is 54 cm². A larger, similar cube has sides twice as long. Find its surface area.
▶️ Answer/Explanation
Scale factor \( k = 2 \)
Surface area ratio = \( k^2 = 4 \)
New surface area = \( 4 \times 54 = \boxed{216 \text{ cm}^2} \)
Example:
Two similar cones have volumes in the ratio 27:125. Find the scale factor of enlargement.
▶️ Answer/Explanation
Volume ratio = \( k^3 = 27:125 \Rightarrow k = \sqrt[3]{27}:\sqrt[3]{125} = 3:5 \)
Scale factor = \( \boxed{3:5} \)
Similarity in Triangles
Similarity in Triangles
Two shapes are similar if they have the same shape but not necessarily the same size. Their corresponding angles are equal, and their corresponding sides are in the same ratio (scale factor).
To Show Two Triangles Are Similar:
- AAA (Angle-Angle-Angle): If all three angles are equal, the triangles are similar.
- SAS (Side-Angle-Side): If two sides are in the same ratio and the included angle is equal.
- SSS (Side-Side-Side): If all three sides are in the same ratio.
Once triangles are known to be similar, we can use scale factor to:
- Find unknown sides or angles
- Give geometric reasons for similarity
Example:
Triangles \( ABC \) and \( DEF \) are such that \( \angle A = \angle D \), \( \angle B = \angle E \), and \( \angle C = \angle F \). Are the triangles similar?
▶️ Answer/Explanation
All corresponding angles are equal.
By AAA criterion, \( \triangle ABC \sim \triangle DEF \)
Reason: All angles equal → triangles are similar (AAA).
Example:
In the diagram, \( DE \parallel BC \). Prove that \( \triangle ADE \sim \triangle ABC \).
▶️ Answer/Explanation
- \( DE \parallel BC \Rightarrow \angle ADE = \angle ABC \) (corresponding angles)
- \( \angle DAE = \angle CAB \) (common angle)
So, \( \triangle ADE \sim \triangle ABC \) by AAA similarity.
Reason: Two corresponding angles equal → triangles are similar (AAA).
Example:
Triangles \( PQR \) and \( XYZ \) have sides:
\( PQ = 4 \), \( QR = 6 \), \( PR = 5 \)
\( XY = 6 \), \( YZ = 9 \), \( XZ = 7.5 \)
Are the triangles similar?
▶️ Answer/Explanation
- Check side ratios:
\( \frac{PQ}{XY} = \frac{4}{6} = \frac{2}{3} \)
\( \frac{QR}{YZ} = \frac{6}{9} = \frac{2}{3} \)
\( \frac{PR}{XZ} = \frac{5}{7.5} = \frac{2}{3} \) - All sides are in the same ratio.
So, \( \triangle PQR \sim \triangle XYZ \) by SSS similarity.
Example :
Triangles \( LMN \) and \( PQR \) are similar.
\( LM = 6 \), \( MN = 9 \), \( LN = 7.5 \)
\( PQ = 10 \), find the corresponding length \( QR \).
▶️ Answer/Explanation
Let’s find the scale factor:
\( \frac{PQ}{LM} = \frac{10}{6} = \frac{5}{3} \)
Apply same scale to other sides:
\( QR = \frac{5}{3} \times 9 = \boxed{15} \)