Similar shapes

Similar shapes are the same shape but can be different sizes, like a photo and its enlargement. Find the scale factor, use it to work out missing sides, and learn the conditions that make two triangles similar.

How can you find the height of a tree without climbing it? Its shadow and your own shadow make two triangles of the same shape.

Two shapes are similar when one is an enlargement of the other: the same shape, maybe a different size. What changes when a shape is enlarged, and what stays the same?

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

Double the sides

In an enlargement, every length is multiplied by the same scale factor $k$ , and every angle stays the same. Congruent shapes are similar with $k = 1$ .

k = \frac{\text{copy length}}{\text{matching original length}}

Triangles are similar when any one of these holds:

Condition	What matches
AA	two angles equal (so the third is too)
SSS	all three pairs of sides in the same ratio
SAS	two pairs of sides in the same ratio, and the angle between them equal

Translations, reflections and rotations move a shape without changing its size, so they make congruent shapes. An enlargement makes a similar shape.

A triangle has sides 3 cm, 4 cm and 5 cm. A copy has every side doubled. What happens to its angles?They stay the sameThey halveThey double

TryTry it on the board and compare the result with your prediction.

Scale a triangle

Then try a scale factor less than 1. What happens to the copy?

Try it yourselfYour edits are saved in this browser.

Make a copy with scale factor 2. Divide each copy side by its matching original side. Did the angles change, as you predicted?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Scale factors

Here is the routine.

Worked exampleBefore each step, say what you think comes next.

$\text{original } 2, 3, 4;\ \text{copy shortest} = 5$ Match the sides: shortest with shortest.
$k = 5 \div 2 = 2.5$ Scale factor: copy ÷ original.
$4 \times 2.5 = 10$ Multiply the original side by k.
$10 \text{ cm}$ Check: 10 ÷ 4 = 2.5 as well.

Read Mia's working. Does a photo stay the same shape when you add?

A triangle has sides 4 cm, 6 cm and 8 cm. A similar triangle's shortest side is 6 cm. How long is its longest side, in cm?

Mia enlarges a photo. Click the line where her working goes wrong.

Two triangles both have angles of 40° and 75°. Are they similar?Yes, by AA: the third angles are both 65°No: you need the sidesOnly if they are the same sizeYou cannot tell

ModifyChange the problem on purpose and see what happens.

Rectangles and shadows

Need a hint?

Multiply both the width and the height by the same number.

Try it yourselfYour edits are saved in this browser.

Make the second rectangle similar to the 4 by 2 rectangle, but not congruent.

Use the board. This ticks when you get there.

Now make a rectangle that is not similar to it. Its angles are all 90°: why is it not similar?

Use the board. This ticks when you get there.

A 1.5 m post casts a 2 m shadow. At the same time, a tree casts a 12 m shadow. How tall is the tree, in metres?

Which is true?All rectangles are similarAll rectangles are congruentAll squares are similar, but not all rectangles areNo two squares are similar

PractiseA few on your own, to check you have it.

Similar shapes

A triangle's sides are 5, 7 and 9 cm. A similar triangle's matching sides are 15, 21 and 27 cm. What is the scale factor?

A 10 cm by 6 cm rectangle is reduced so the 10 cm side becomes 5 cm. How long is the 6 cm side now, in cm?

A triangle is moved by a transformation, and its image is similar but not congruent. Which transformation was it?A rotationA translationA reflectionAn enlargement

Review: two triangles have equal angles of 50° and 60°, and the sides between those angles are both 8 cm. Are they congruent?No, they are only similarYes, by ASAYes, by SSSYou cannot tell

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

Find the scale factor

Then say what the copy's other two sides are, and how you know the angles match.

Try it yourselfYour edits are saved in this browser.

Need a hint?

Which original side matches y?

Another hint

k = the copy side ÷ the matching original side.

The original's sides are 6, 8 and 10 cm. In the copy, side y (matching the 8 cm side) is 12 cm. Set the scale factor.

Use the board. This ticks when you get there.

For teachers

Students scale a triangle by a scale factor, find missing sides of similar shapes, and compare rectangles that are similar, congruent or neither (AC9M8SP01, nodes 2 and 3). About 40 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can find a scale factor from a pair of matching sides.
I can use a scale factor to find a missing side.
I can say whether two shapes are congruent, similar or neither, and why.

Before class

Do the Run board yourself first. Students need congruent triangles (last lesson) and multiplying by decimals.

The one idea: similar means every length times the same scale factor, with the angles unchanged. The common mistake is adding a fixed amount to every side (Mia's mistake).

Similarity conditions for triangles: AA, SSS (all sides in one ratio) and SAS (two sides in one ratio, the included angle equal). Only AA is practised much here; SSS appears in Practise. The link to transformations: translations, reflections and rotations give congruent shapes; an enlargement gives a similar shape.

Wider: measure something tall outside with shadows and a metre ruler.

Running the lesson

Predict: ask what doubling does to a corner's angle.
Run: ask what each copy side ÷ original side gives.
Investigate: ask Mia whether her photo would look stretched.
Modify: ask why two rectangles with all 90° angles may not be similar.
Make: ask which original side matches y.

The hard part (on purpose)

Matching the sides. Ask students to say which side matches which (shortest with shortest) before they divide.

When students get stuck

A student adds the same amount to every side. Ask: Would the picture look the same shape? Fix: Find k by dividing, then multiply every side by k.
A student divides the wrong way (original ÷ copy). Ask: Is the copy bigger or smaller? Should k be more or less than 1? Fix: k = copy ÷ original.
A student says all rectangles are similar because the angles match. Ask: Is a 4 by 2 rectangle the same shape as a 4 by 3? Fix: For shapes other than triangles, check the sides' ratios too.

Stretch and support

Stretch: A similar triangle has scale factor 3. How many times bigger is its area? Test it with the 3, 4, 5 triangle.

Support: A table with original sides in one column and copy sides in the next, with a ÷ column for k.

Where it leads

This covers AC9M8SP01 nodes 2 and 3.

Where it leads: proving quadrilateral properties with congruent triangles (next lesson), congruence algorithms, and trigonometry in Year 10 (which rests on similar right-angled triangles).

What each step is for

double the sides: Committing to what happens to the angles when sides double shows whether they confuse lengths with angles.
scale a triangle: Changing the scale factor shows every side multiplied by the same number and the shape unchanged.
scale factors: Find k from a matching pair and multiply; adding a fixed amount is the slip (Mia); AA is enough because the third angle follows.
rectangles and shadows: Rectangles can have equal angles and still not be similar; similar triangles measure a tree from shadows; all squares are similar but not all rectangles.
similar shapes: Finding k, missing sides and reductions, mixed; adding instead of multiplying is the planted slip; transformations link congruence and similarity; the congruence review links the two ideas.
find the scale factor: Working back from one copy side to k shows they can find a scale factor and use it.

Now you can

Find a scale factor from a pair of matching sides.
Use a scale factor to find a missing side.
Say whether two shapes are congruent, similar or neither, and why.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson18. Proving quadrilateral propertiesA diagonal cuts a quadrilateral into two triangles. When those triangles are congruent, their matching sides and angles are equal, and that proves a property for every shape of the kind: a rectangle's diagonals are equal, a parallelogram's cut each other in half, a kite's cross at 90°.

Source: content/maths/angles/similar-shapes/lesson.md