Similar shapes

The plane and space, grade 11 (3 units) Lesson 5 of 15Learning

Similar shapes

Two shapes are similar when one of them is an enlargement or a reduction of the other.

Every length grows or shrinks by the same number, $\displaystyle k$ – the scale factor.
The angles do not change, so the shape stays the same shape.
A map and a model are similar to the original, and their scale factor is the scale.

Enlarging a photo

A photo 10 cm by 15 cm is enlarged to a photo 20 cm by 30 cm:

$\displaystyle {\frac{20}{10}}={2}$ and $\displaystyle {\frac{30}{15}}={2}$ : both lengths grew 2 times, so the photos are similar, and $\displaystyle {k}={2}$ .
A photo 20 cm by 25 cm is not similar to it: $\displaystyle {\frac{20}{10}}={2}$ , but $\displaystyle \frac{25}{15}\approx1.67$ .
A photo printed on such a sheet without cropping is stretched less along its length than across, and the faces in it look too wide.

Exercise 1

A photo 9 cm by 13 cm is to be enlarged without cropping or distorting it.
Which of the frames fit it: 18 cm by 26 cm, 27 cm by 39 cm, 20 cm by 30 cm?
Explain.

Exercise 2

A photo 6 cm by 9 cm is enlarged so that its 6 cm side becomes 15 cm.
How many centimetres will its other side be?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The area of an enlargement

An enlargement $\displaystyle k$ times makes the area $\displaystyle k^2$ times as large: the length grows $\displaystyle k$ times, and so does the width, and the area is their product.

The 10 by 15 photo: $\displaystyle 150$ cm². The 20 by 30 photo: $\displaystyle 600$ cm².
$\displaystyle {600\div150}={4}={2^2}$ .
The same holds for any two similar shapes – triangles and discs too: every small square of the shape grows $\displaystyle k$ times along and $\displaystyle k$ times across.

When $\displaystyle S_1$ and $\displaystyle S_2$ are the areas of two similar shapes, and the scale factor from the first to the second is $\displaystyle k$ :

\displaystyle \frac{S_2}{S_1}

\displaystyle =

\displaystyle k^{2}

Example: two pizzas

At a pizzeria, a pizza 30 cm across costs 40 ₪, and a pizza 40 cm across costs 60 ₪.
Which is better value?

The solution below compares the ratio of the areas with the ratio of the prices.

Any two discs are similar, and the scale factor is the ratio of the diameters: $\displaystyle {k}={\frac{40}{30}}={\frac{4}{3}}$ .
The ratio of the areas: $\displaystyle {k^2}={\frac{16}{9}\approx1.78}$ – the large pizza holds $\displaystyle 1.78$ times as much pizza.
The price grows only $\displaystyle {\frac{60}{40}}={1.5}$ times, so the large pizza is better value.

Exercise 3

A poster is a 3 times enlargement of a photo 20 cm by 30 cm.

The photo’s area, in cm²

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The poster’s area, in cm²

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

An A3 sheet is similar to an A4 sheet, and its area is twice the area of an A4 sheet.
What is the scale factor from A4 to A3?
Round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

Two flower beds are similar, and the scale factor from the small one to the large one is 1.5.
The small bed needs 4 kg of fertiliser.
How many kilograms does the large bed need?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: the plane and space, grade 11 (3 units)

Open as a page

Scale

The real length – $\displaystyle l$ the length on the map, at a scale of $\displaystyle 1:n$
$\displaystyle L$ $\displaystyle =$ $\displaystyle n\cdot l$
The real area – $\displaystyle s$ the area on the map, at a scale of $\displaystyle 1:n$
$\displaystyle S$ $\displaystyle =$ $\displaystyle n^{2}\cdot s$

Similar shapes

The scale factor – $\displaystyle a_1$ , $\displaystyle a_2$ corresponding sides
$\displaystyle k$ $\displaystyle =$ $\displaystyle \frac{a_2}{a_1}$
The ratio of the areas of similar shapes
$\displaystyle \frac{S_2}{S_1}$ $\displaystyle =$ $\displaystyle k^{2}$
A height by its shadow – $\displaystyle H$ , $\displaystyle S$ the object and its shadow; $\displaystyle h$ , $\displaystyle s$ a stick and its shadow
$\displaystyle \frac{H}{S}$ $\displaystyle =$ $\displaystyle \frac{h}{s}$

Trigonometry in the right-angled triangle

Sine – $\displaystyle a$ the side opposite the angle, $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
Cosine – $\displaystyle b$ the side adjacent to the angle
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
Tangent – the opposite side over the adjacent side
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The angle from its tangent (and likewise $\displaystyle \sin^{-1}$ , $\displaystyle \cos^{-1}$ )
$\displaystyle \alpha$ $\displaystyle =$ $\displaystyle \tan^{-1}\frac{a}{b}$
Pythagoras’ theorem
$\displaystyle a^2+b^2$ $\displaystyle =$ $\displaystyle c^2$
A gradient in percent – $\displaystyle \alpha$ the angle with the horizontal
$\displaystyle p$ $\displaystyle =$ $\displaystyle 100\cdot\tan\alpha$