Similar shapes (Edexcel GCSE Maths): Revision Notes
Similar shapes
What are similar shapes?
Two shapes are similar when they have exactly the same shape but different sizes. In similar shapes, corresponding angles are equal and corresponding sides are in the same ratio (proportional).
For triangles to be similar, they must satisfy one of three specific conditions that we'll explore in detail.
Three conditions for similar triangles
Condition 1: All angles are equal (AAA)
When all three pairs of corresponding angles in two triangles are equal, the triangles are similar.
- Each angle in the first triangle equals its corresponding angle in the second triangle
- You only need to check two angles, as the third will automatically be equal
- The triangles will be the same shape but different sizes
Remember: You only need to prove two pairs of angles are equal - the third pair will automatically be equal since angles in a triangle sum to 180°.
Condition 2: All sides are proportional (SSS)
When all three pairs of corresponding sides are in the same ratio, the triangles are similar.
- Compare each side of one triangle to its corresponding side in the other triangle
- All ratios must be equal
- For example, if the ratios are 3:6, 4:8, and 7:14, they all equal 1:2
Condition 3: Two sides proportional with equal included angle (SAS)
When two pairs of corresponding sides are in the same ratio and the angle between them is equal, the triangles are similar.
- The included angle is the angle formed between the two sides you're comparing
- Both side ratios must be equal
- The angle between these sides must be the same in both triangles
Finding missing lengths in similar triangles
When triangles are similar, you can find unknown lengths using proportional ratios. The process involves setting up equations with corresponding sides and solving for the unknown value.
Worked Example: Step-by-Step Method
Step 1: Identify corresponding sides - match sides that are in the same position in each triangle
Step 2: Set up the ratio - write corresponding sides as equal fractions
Step 3: Substitute known values - put in the measurements you know
Step 4: Solve for the unknown - cross multiply and calculate
Calculation process:
- Write the ratio: corresponding side 1 ÷ corresponding side 2 = corresponding side 3 ÷ corresponding side 4
- Always put the unknown length on top of the fraction
- Cross multiply to solve: if a/b = c/d, then a = (c × b) ÷ d
Key facts about similar shapes
Corresponding angles:
- Angles in matching positions are always equal in similar shapes
- This is true regardless of the size difference between the shapes
Corresponding sides:
- Sides in matching positions are always in the same ratio
- This ratio is called the scale factor between the shapes
- All corresponding side pairs have the same scale factor
The scale factor tells you how many times larger (or smaller) one shape is compared to another. If the scale factor is 2:1, then the first shape is twice as large as the second.
Identifying similar triangles
Look for these features when spotting similar triangles:
- Equal angles - triangles may be rotated or flipped but angles remain the same
- Proportional sides - even if triangles are different sizes, corresponding sides maintain the same ratio
- Same shape - the triangles have identical shapes despite size differences
Triangle pairs can be similar even when:
- One triangle is larger than the other
- The triangles are oriented differently (rotated or reflected)
- The triangles are positioned differently on the page
Don't be fooled by orientation! Similar triangles can be rotated, flipped, or positioned differently, but they will still maintain their angle and ratio relationships.
Exam tips for similar shapes
These strategies will help you tackle similar triangle questions effectively in your exams.
Essential Exam Strategies:
- Start with angles - often the easiest way to spot similarity
- Write ratios carefully - make sure you match corresponding sides correctly
- Show your working - set up the ratio equation clearly before calculating
- Check your answer - verify that your calculated length makes sense with the scale factor
Key Points to Remember:
- Two triangles are similar when they meet one of three conditions: all angles equal, all sides proportional, or two sides proportional with equal included angle
- Corresponding sides in similar triangles are always in the same ratio - this ratio is the scale factor
- Corresponding angles in similar shapes are always equal regardless of size difference
- To find missing lengths, set up proportional ratios using corresponding sides and solve by cross multiplication
- Similar triangles can be rotated or flipped but still maintain their angle and ratio relationships