Syllabus Edition
First teaching 2021
Last exams 2024
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Revision Note
Test Yourself
Author
Amber
Expertise
Maths
Similarity
What are similar shapes?
- Two shapes are similar if they have the sameshape and their corresponding sidesare inproportion
- One shape is anenlargement of the other
- If two triangles of different sizeshave the same angles they are similar
- Other shapes can have the same angles and not be similar
How do we prove that two shapes are similar?
- To show that two non triangular shapes are similar you need to show that their corresponding sides are in proportion
- Divide the length of one side by the length of the same side on the other shape to find thescale factor
- If the scale factor is the same for all corresponding sides, then the shapes are similar
- If one shape can be shown to be an enlargement of the other, then the two shapes are similar
How do we prove that two triangles are similar?
- To show that two triangles are similar you simply need to show that their angles are the same
- This can be done through angle properties, look for isosceles triangles, vertically opposite angles and angles on parallel lines
- The triangles may not look similar and may be facing in different directions to each other, so concentrate on finding the angles
- it may help to sketch both triangles next to each other and facing the same direction
- If a question asks you to prove two triangles are similar, you will need to state that corresponding angles in similar triangles are the same and you will need to give a reason for each corresponding equal angle
- The triangles can often be opposite each other in an hourglass formation, look out for the vertically opposite, equal angles
Exam Tip
- Proving two shapes are similar can require a lot of writing, you do not need to write in full sentences, but you must make sure you quote all of the keywords to get the marks
Worked example
a)
Prove that the two rectangles shown in the diagram below are similar.
Use the two lengths (15 cm and 6 cm) to find the scale factor.
Multiply this by the width of the smaller rectangle to see if it applies to the width as well.
The two rectangles are similar, with a scale factor of 2.5
b)
In the diagram below, AB and CD are parallel lines.
Show that triangles ABX and CDX are similar.
State the equal angles by name, along with clear reasons.
Don't forget to state that similar triangles need to have equal corresponding angles.
Angle AXB = angle CXD (Vertically opposite angles are equal) All three corresponding angles are equal, so the two triangles are similar
Angle ABC = angle BCD (Alternate angles on parallel lines are equal)
Angle BAD = angle ADC (Alternate angles on parallel lines are equal)
Similar Lengths
How do I work with similar lengths?
- Equivalent lengths in two similar shapes will be in the same ratio and are linked by a scale factor
- Normally the first step is to find this scale factor
- STEP 1
Identify equivalent known lengths - STEP 2
Establish direction- If the scale factor is greater than 1 the shape is getting bigger
- If the scale factor is less than 1 the shape is getting smaller
- STEP 3
Find the scale factor- Second Length ÷ First Length
- STEP 4
Use scale factor to find the length you need
- STEP 4
Exam Tip
- If similar shapes overlap on the diagram (or are not clear) draw them separately
- For example, in this diagram the triangles ABC and APQ are similar:
So we would redraw them separately before we start:
Worked example
ABCDandPQRSare similar shapes.Find the length ofPS.
As the two shapes are mathematically similar, there will exist a value of ksuch that and
.
is known as the scale factor.
Form an equation using the two known corresponding sides of the triangle.
Solve to find .
Substitute into.
Solve to find .
Similar Areas & Volumes
What are similar shapes?
- Two shapes are mathematically similar if one is an enlargement of the other
- If two similar shapes are linked by thescale factor, k
- Equivalent areas are linked by an area factor, k2
- Equivalent volumes are linked by a volume factor, k3
How do I work with similar shapes involving area or volume?
- STEP 1
Identify the equivalent known quantities- These could be for lengths, areas or volumes
- STEP 2
Establish direction- Are they getting bigger or smaller?
- STEP 3
Find the Scale Factor from two knownlengths, areasorvolumes- Second Quantity ÷ First Quantity
- Check the scale factor is > 1 if getting bigger and < 1 if getting smaller
- If the scale factor, s.f., is from two lengths, write it as k= s.f.
- If the scale factor, s.f., is from two areas, write it as k2= s.f.
- If the scale factor, s.f., is from two volumes, write it as k3= s.f.
- STEP 4
Use the value of the scale factor you have found to convert other corresponding lengths, areas or volumes using- Area Scale Factor = (Length Scale Factor)2
- Or Length Scale Factor = √(Area Scale Factor)
- Volume Scale Factor = (Length Scale Factor)3
- Or Length Scale Factor = ∛(Volume Length Factor)
- Area Scale Factor = (Length Scale Factor)2
- Use the scale factor to find a new quantity
Exam Tip
- Take extra care not to mix up which shape is which when you have started carrying out the calculations
- It can help to label the shapes and always write an equation
- For example if shape A is similar to shape B:
- length A = k(length B)
- areaA = k2(area B)
- volume A = k3(volume B)
- For example if shape A is similar to shape B:
Worked example
SolidAand solidBare mathematically similar.
The volume of solidAis 32 cm3.
The volume of solid B is 108 cm3.
The height of solid Ais 10 cm.
Find the height of solidB.
Calculate , the scale factor of enlargement for the volumes, using
,
Or.
For similar shapes, if the volume scale factor is , then the length scale factor is
.
FInd .
Substitute into formula for the heights of the similar shapes. ,
Height ofB = 15 cm
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