Geometry II

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Welcome to Class !!

We are eager to have you join us !!

In today’s Mathematics class, We will continue our lesson on Geometry. We hope you enjoy the class!

 

 

 

GEOMETRY CONT’D

Areas and Volumes of similar shaped

Scale Factor (Length Ratio)

P          9cm              Q

A                  B

 

2                                         6cm

 

D    3          C

S                                    R

 

$Scale factor = \frac{9}{3} or \frac{6}{2} = \frac{3}{1}$

Area factor   = (Scale factor)²

${\left(\frac{3}{1}\right)}^2 = \frac{9}{1} or 9.1$

 

If the ratio of the length is X: Y

Then the ratio of the area is  X²: Y²

 

Example 1:

In the diagram below, if the area of triangle ABC is 48cm², find the area of triangle XYZ to 3s.f

A                                         X

 

 

55⁰

 

 

65⁰       60⁰                                                    60⁰

                  B        12cm         C            Y      4cm       Z

Solution

In ∆ABC,  <A   = 180 – (65 + 60)  = 55

In ∆XYZ, <Y  = 180 – (55-60) = 65⁰

Since then corresponding angles are equal, ∆ABC and ∆XYZ are similar

$$\begin{gathered} Scale factor = \frac{4}{12} = \frac{1}{3} \\ Area factor = {\left(\frac{1}{3}\right)}^2 = \frac{1}{9} \\ \frac{Area of ∆XYZ}{Area of ∆ABC} = \frac{1}{9} \\ \frac{Area of ∆XYZ}{48} = \frac{1}{9} \\ Area of ∆XYZ \times 9 = 48 \\ Area of ∆XYZ = \frac{48}{9} \\ = 5.3333 c{m}^2 \\ Area of ∆XYZ = 5.3 c{m}^2 to 2 s.f \end{gathered}$$

 

 

The volume of Similar Shapes

Volume factor = (Scale factor)³

If the ratio of the length is X: Y

Then the ratio of the volume is X³: Y³

 

Example 2: The ratio of a cylinder of volume 2970cm³ is 30mm. find the volume of a similar cylinder of radius 40mm.

 

Solution

$$\begin{gathered} Scale factor = \frac{40mm}{30mm} = \frac{4}{3} \\ Volume ratio = {\left(\frac{4}{3}\right)}^3 = \frac{4^3}{3^3} \\ \frac{V}{2970} = \frac{4^3}{3^3} \\ Cross multiply \\ V \times 27 = 2970 \times 64 \\ V = \frac{2970 \times 64}{27} = 7040 \\ The volume of the cylinder is 7040 c{m}^3 \end{gathered}$$

 

 

 

 

We have come to the end of this class. We do hope you enjoyed the class?

Should you have any further question, feel free to ask in the comment section below and trust us to respond as soon as possible.

In our next class, we will be talking about the Area of Plane Figures. We are very much eager to meet you there.

 

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