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19.3 Using Similarity

GCSE Maths Foundation · Edexcel · Chapter 19: Congruence, Similarity and Vectors

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: solving real-life problems with similar shapes and triangles. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Use similar triangles to find heights, for example from shadows
  • Decide whether real-life shapes such as photos, screens and frames are similar
  • Use the fact that perimeters scale by the same factor as lengths
  • Form and solve an equation to find a missing part of a side

Key words

  • Similar: Same shape, different size. Matching angles are equal and every length is multiplied by the same scale factor.
  • Ratio of sides: Long side ÷ short side. Similar rectangles have the same ratio of sides.
  • Perimeter: The total distance around a shape. In similar shapes it scales like any other length.

Heights from shadows

At the same moment the sun's rays hit everything at the same angle. A vertical object and its shadow make a right-angled triangle, so all these triangles are similar.

Worked example: height of a building

Scale factor = building shadow ÷ stick shadow = 18 ÷ 2 = 9
Height h = 1.5 × 9 = 13.5 m

Answer: 13.5 m

Are real-life shapes similar?

For rectangles, compare long side ÷ short side. If the two answers are equal, the rectangles are similar.

Worked example: enlarging a photo

A photo is 12 cm by 18 cm. It is enlarged so the short side is 30 cm.
Scale factor = 30 ÷ 12 = 2.5. Long side = 18 × 2.5 = 45 cm

Answer: 30 cm by 45 cm

Worked example: a border breaks similarity

Photo: 15 ÷ 10 = 1.5. Frame with border: 19 ÷ 14 = 1.357…
The ratios are different, so the frame is not similar to the photo.

Answer: Not similar

Key fact: Adding the same amount to every side does not give a similar shape. Only multiplying every side by the same number does.

Perimeters of similar shapes

If every side is multiplied by k, the perimeter is also multiplied by k. The angles do not change.

Worked example: using perimeters

Two similar triangles have perimeters 24 cm and 60 cm. The shortest side of the small triangle is 5 cm.
Scale factor = 60 ÷ 24 = 2.5. Shortest side of the large triangle = 5 × 2.5 = 12.5 cm

Answer: 12.5 cm

Finding part of a side

Sometimes the unknown is only part of a side. Find the scale factor from sides you know, then write an equation.

Worked example: find x

Scale factor from ADE to ABC = BC ÷ DE = 10 ÷ 4 = 2.5
So AB = 2.5 × AD: x + 6 = 2.5x
6 = 1.5x, so x = 4 cm. Check: AB = 10 and 10 ÷ 4 = 2.5 ✓

Answer: x = 4 cm

Common mistakes

  • Mixing up which shadow goes with which object. Write the two matching pairs side by side before dividing.
  • Thinking a frame with an even border is similar to the photo inside it.
  • Using the part of a side (DB) instead of the whole side (AB) of the big triangle.
  • Forgetting to change units, e.g. a model in cm and a real object in m.

Revision questions

Try these, then check your answers below each one.

Question 1. A 1.2 m pole casts a shadow 0.8 m long. At the same time a building casts a shadow 14 m long. How tall is the building?

Answer: 21 m

Working: Scale factor = 14 ÷ 0.8 = 17.5. Height = 1.2 × 17.5 = 21 m.

Question 2. A photo is 8 cm by 12 cm. It has a 1 cm border all the way round, making a frame 10 cm by 14 cm. Ben says the frame is similar to the photo. Is Ben correct?

Answer: No

Working: Photo: 12 ÷ 8 = 1.5. Frame: 14 ÷ 10 = 1.4. Different ratios, so they are not similar.

Question 3. Two triangles are similar. Their perimeters are 15 cm and 36 cm. A side of the small triangle is 4 cm. Work out the matching side of the large triangle.

Answer: 9.6 cm

Working: Scale factor = 36 ÷ 15 = 2.4. Side = 4 × 2.4 = 9.6 cm.

Related lessons

Continue in the full course

This lesson is section 19.3 (EGCMF1903) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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