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Year 10
Higher

Problem solving with advanced similarity knowledge

I can use my knowledge of similarity to solve problems.

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New
New
Year 10
Higher

Problem solving with advanced similarity knowledge

I can use my knowledge of similarity to solve problems.

Download all resources
Share activities with pupils

Lesson details

Key learning points

  1. A scaled version of a shape/object is very useful.
  2. Being able to scale the perimeter/area/volume can reduce the number of calculations.
  3. This knowledge of similarity can be extended to include surface area.

Common misconception

"One shape has an area that is 21% larger than another shape. I can't find the area scale factor between the shapes, since I don't know the actual area of either shape."

It is possible to find scale factors, even if only a percentage that describes their areas is given. One shape will have an area of 100%, so the other shape will have an area 21% larger, at 121%. The scale factor is 1.21 because 100% × 1.21 = 121%.

Keywords

  • Similar - Two shapes are similar if the only difference between them is their size. Their side lengths are in the same proportions.

  • Invariant - A property of a shape is invariant if that property has not changed after the shape is transformed.

  • Enlargement - Enlargement is a transformation that causes a change of size.

  • Scale factor - A scale factor is the multiplier between similar shapes that describes how large one shape is compared to the other.

  • Volume - The amount of space occupied by a closed 3D shape.

Even on questions that do not explicitly give a scale factor table, it can be helpful to encourage students to construct their own. This is especially helpful when given lengths, areas, and volumes, and the linear, area, and volume scale factors are needed.
Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Starter quiz

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6 Questions

Q1.
theorem states that the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of the hypotenuse.
Correct Answer: Pythagoras', Pythagoras, Pythagoras's
Q2.
The lengths of the 3 edges of some triangles are given. Select all the right-angled triangles.
Correct answer: 8 cm, 15 cm, 17 cm
8 cm, 10 cm, 12 cm
15 cm, 20 cm, 25 cm
9 cm, 15 cm, 18 cm
Correct answer: 9 cm, 12 cm, 15 cm
Q3.
A right-angled triangle has a hypotenuse of 25 m. Select the possible lengths of the two shorter sides.
12 m and 18 m
Correct answer: 15 m and 20 m
Correct answer: 7 m and 24 m
8 m and 20 m
10 m and 15 m
Q4.
Which of these pairs of triangles are congruent?
Correct Answer: An image in a quiz
An image in a quiz
An image in a quiz
An image in a quiz
An image in a quiz
Q5.
Triangle ABC and triangle DEF are congruent and AB > BC. The length of the side $$x$$ is cm.
An image in a quiz
Correct Answer: 8
Q6.
Match each letter which the correct statement to complete the proof that triangle DAC and triangle ABC are congruent.
An image in a quiz
Correct Answer:a,∠ABC

∠ABC

Correct Answer:b,AC

AC

Correct Answer:c,DC

DC

Correct Answer:d,RHS

RHS

Exit quiz

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6 Questions

Q1.
Two shapes are if the only difference between them is their size. Their side lengths are in the same proportions.
Correct Answer: similar
Q2.
The multiplier between the surface areas of a pair of similar objects is $$k^2$$. Select the correct statements.
Correct answer: The multiplier between their corresponding edge lengths is $$k$$
The multiplier between their corresponding edge lengths is $$\frac {1}{2} k$$
The multiplier between their volumes is $$3k$$
Correct answer: The multiplier between their volumes is $$k^3$$
The multiplier between their volumes is $$\sqrt k$$
Q3.
Shapes X and Y are similar to each other. The volume of shape Y is 216 times larger than the volume of shape X. The linear scale factor from X to Y is .
Correct Answer: 6, six
Q4.
Shapes X and Y are similar. The perimeter of shape Y is 125% of the perimeter of shape X. Work out the area scale factor from X to Y.
$$\frac{5}{4}$$
$$\frac{4}{5}$$
$$\frac{16}{25}$$
Correct answer: $$\frac{25}{16}$$
Q5.
Pyramid A and pyramid B are similar. The surface area of pyramid A is 50 cm² and its volume is 120 cm³. The surface area of pyramid B is 450 cm². Calculate the volume of pyramid B.
An image in a quiz
3480 cm³
Correct answer: 3240 cm³
1920 cm³
1080 cm³
Q6.
Cuboid A and cuboid B are similar. The volume of cuboid B is 7680 cm³. The surface area of B is cm².
An image in a quiz
Correct Answer: 2528