3.8.2 Similarity
3.8.2 Similarity
3.8.2 Similarity
Similarity refers to the degree to which two things or entities resemble each other. It can refer to a wide range of things, including the similarity between two people, two objects, two ideas, or two sets of data.
Similarity is often measured or evaluated based on certain characteristics or features of the things being compared.
For example, two people may be considered similar if they share common interests, values, or physical traits.
Two objects may be considered similar if they have similar shapes, colors, or functions. In data analysis, similarity can be measured using various metrics, such as Euclidean distance, cosine similarity, or Jaccard similarity, depending on the type of data and the application.
To prove that two shapes are similar, you need to demonstrate that they have the same shape but possibly different sizes.
Two shapes are similar if their corresponding angles are congruent (have the same measure) and their corresponding sides are proportional (have lengths in the same ratio).
Here are the steps to prove that two shapes are similar:
Identify the corresponding angles in each shape.
Corresponding angles are angles that occupy the same relative position in each shape
Measure the corresponding angles and verify that they are congruent.
If the corresponding angles have the same measure, then the shapes have the same shape.
Identify the corresponding sides in each shape.
Corresponding sides are sides that are opposite or adjacent to corresponding angles
Measure the corresponding sides and verify that they are proportional.
If the ratios of the corresponding side lengths are the same, then the shapes are similar.
Write a similarity statement.
Use the corresponding angles and sides to write a statement that shows the similarity between the two shapes.
The statement should include the name of each shape and the similarity ratio, which is the ratio of corresponding side lengths.
For example, if you have two triangles,\( ABC \) and \( PQR \), and you want to prove that they are similar, you would follow these steps:
Identify the corresponding angles:
angle \( A \) corresponds to angle \( P \), angle \( B \) corresponds to angle \( Q \), and angle \( C \) corresponds to angle \( R \).
Measure the corresponding angles and verify that they are congruent:
angle \( A = \) angle \( P \), angle \( B = \) angle \( Q \), and angle \( C = \) angle \( R \).
Identify the corresponding sides:
side \( AB \) corresponds to side \( PQ \), side \( BC \) corresponds to side \( QR \), and side \( AC \) corresponds to side \( PR \).
Measure the corresponding sides and verify that they are proportional:
\( \frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR} \)
Write a similarity statement:
“Triangle \( ABC \) is similar to triangle \( PQR \) with a similarity ratio of \( \frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR} \)”.
Worked example:
In the diagram, \( AB \) and \( CD \) are parallel.
\( AD \) and \( BC \) intersect at right angles at the point \( X \).
\( AB=10 \) cm, \( CD=5 \) cm, \( AX=8 \) cm and \( BX=6 \) cm.
Use similar triangles to calculate \( DX \).
Alternate angles in parallel lines are equal
Find the scale factor of enlargement by dividing \( AB \) by \( CD \)
Scale factor\( =\frac{10}{5}=2 \)
Find the corresponding side of \( DX \)
\( DX \) corresponds to \( AX \)
\( \frac{AX}{DX}=2 \)
\( DX=\frac{AX}{2} \)
\( DX=\frac{8}{2} \)
\( DX=4 \)
\( DX=4 cm \)
Similar Areas and Volumes:
When two shapes are similar, their areas and volumes are also proportional to the square of their corresponding sides. This means that if the ratio of corresponding side lengths between two similar shapes is k, then the ratio of their areas is k2, and the ratio of their volumes is k3.
For example, if you have two similar triangles, and the ratio of their corresponding side lengths is \( 2 \) : \( 1 \), then the ratio of their areas is \( (2 :1)^2=4:1 \). This means that the larger triangle has four times the area of the smaller triangle.
Similarly, if you have two similar spheres, and the ratio of their corresponding diameters is \( 3:2 \), then the ratio of their volumes is \( (3:2)^3 = 27:8 \). This means that the larger sphere has \( 27 \) times the volume of the smaller sphere.
The same concept applies to other similar shapes, such as rectangles, cylinders, and cones.
In general, when two shapes are similar, their areas and volumes are proportional to the square and cube of their corresponding side lengths, respectively.
This property is very useful in many applications, such as scaling of objects or structures, or in geometry and physics calculations.
Worked example:
Triangle \( ABC \) is mathematically similar to triangle \( PQR \).
The area of triangle \( ABC \) is \( 16 \) cm2.
Calculate the area of triangle \( PQR \).
Find the length scale factor by dividing a length of the larger shape \( (PQ) \) by the corresponding length of the smaller shape \( (AB) \).
\( Length \ scale \ factor\ =\frac{12}{8}=1.5 \)
Square the length scale factor to get the area scale factor
\( Area \ scale \ factor\ =1.5^2=2.25 \)
Multiply the area of smaller shape by the area of scale factor to get the area of larger shape
\( Area \ of \ triangle \ PQR=16\times 2.25=36 \)
\( Area \ of \ triangle \ PQR=36 cm^2 \)
Test yourself
Question 1:
Two cones are mathematically similar.
The total surface area of the smaller cone is \( 80 \) cm2.
The total surface area of the larger cone is \( 180 \)cm2.
The volume of the smaller cone is \( 168 \) cm3.
Calculate the volume of the larger cone.
[3]
Question 2:
\( PQR \) is a triangle.
\( T \) is a point on \( PR \) and \( U \) is a point on \( PQ \).
\( RQ \) is parallel to \( TU \).
Explain why triangle \( PQR \) is similar to triangle \( PU \).
Give a reason for each statement you make.
[3]
\( PT \) : \( TR=4\) : \( 3 \)
Find the ratio \( PU \) : \( PQ \).
[1]
The area of triangle \( PUT \) is \( 20 \) cm2.
Find the area of the quadrilateral \( QRTU \).
[3]
Question 3:
The diagram shows a company logo made from a rectangle and a major sector of a circle.The circle has center \( O \) and radius \( OA \).
\( OA=OD=0.5 \) cm and \( AB=1.5 \) cm.
\( E \) is a point on \( OC \) such that \( OE=0.25 \) cm and angle \( OED=90° \).
Calculate the perimeter of the logo.
[5]
Calculate the area of the logo.
[3]
A mathematically similar logo is drawn.
The area of this logo is \( 77.44 \) cm2.
Calculate the radius of the major sector in this logo.
[3]
Question 4:
The diagram shows two mathematically similar shapes with areas \( 295 \) cm2 and \(159.5 \) cm2.
The width of the larger shape is \( 17 \) cm.
Calculate the width of the smaller shape.
[3]
Question 5:
A solid metal cone has radius \( 10 \) cm and height \( 36 \) cm.
Calculate the volume of this cone.
[The volume, \( V \), of a cone with radius \( r \) and height \( h \) is \( V=\frac{1}{3}r^2h \) ]
[2]