SIMILARITY

SIMILARITYWHAT IS SIMILARITY????Similarity is the name given to two figures having the same shape, but different sizes. In simpler words, one figure is an enlargement of the other. Similarity is the quality of being similar. It refers to the closeness of appearance between two or more objects. It is the relation of sharing properties

DEFINITIONSCorresponding - similar especially in position or purpose; equivalentRatio - the relationship between two numbers or quantities (usually expressed as a quotient)Proportion - the relation between things (or parts of things) with respect to their comparative quantity, magnitude, or degreeCongruent - equilateral, equal, exactly the same (size, shape, etc.)Enlargement - expansion: the act of increasing (something) in size or volume or quantity or scopeTheorem - an idea accepted as a demonstrable truthScale factor reduced ratio of corresponding sides of two similar figures.

EXAMPLESTHESE TWO SHAPES ARE SIMILAR

Two polygons are said to be similar if and only if there is a one-to-one correspondence between their vertices such that:1. Corresponding angles are equal 2. Lengths of corresponding sides are in proportion.EXAMPLE:

BCAD2 cm1 cmWXYZ4 cm2 cm

TRIANGLESCONDITIONS FOR SIMILAR TRIANGLESSide angle side (SAS)

ABC3 cm4 cm 40WYX6 cm8 cm If a pair of corresponding sides of two triangles are in the same proportion and the angle between the sides are equal, then the triangles are similar.

7If the corresponding angles of two triangles are equal, then the two triangles are similar.

ABCYXZAngle angle side triangle (AAS)

. side side side triangles (SSS)In this type of similarities, the triangles are similar when three sides of the triangle are corespnding. So, if all pairs of corresponding sides of two triangles are proportional, then triangles are similar.NOTE: In side side side triangles, their corresponding sides are manified by a certain factor ,K.

6 cm3 cm4 cm ABCWXY6 cm8 cm12 cm

9THE FIGURES WHICH ARE ALWAYS SIMILAR!!! Circles Regular Pentagons Squares Equilateral TrianglesAREAS OF SIMILAR FIGURESThe following rectangles are similar and their ratio of corresponding sides is y. ABCD is of length b and width a.

If two figures are similar and their sides are in the ratio y, then their areas will be in the ratio y2.

YbZYXWyaAabDBC

SIMILARITY IN 3 DIMENSIONAL FIGURES13SIMILAR 3-D FIGURES IN OUR DAILY LIVES

When solid objects are similar, one is an accurate enlargement of the other. Thus, the corresponding sides should be in the same ratio.80 cm48 cm100 cm60 cm140 cm84 cmVOLUMES AND SURFACE AREAS OF SIMILAR 3-D OBJECTSA and B are two similar 3-D shapes. Their ratio of corresponding sides is .

If the ratio of the corresponding sides of two 3-D objects is k, then the ratio of their surface areas is

A a bcBka kbkc

A a bcBka kbkc

If the ratio of the corresponding sides of two 3-D objects is k, then the ratio of their volumes is

When solid objects are similar, one is an accurate enlargement of the other. If two objects are similar and the ratio of corresponding sides (scale factor) is k, then the ratio of their volumes is k3.

A line has one dimension, and the scale factor is used once.

An area has two dimensions, and the scale factor is used twice.

A surface area of 3 dimensional figures also uses the scale factor twice.

A volume has three dimensions, and the scale factor is used three times.

SUMMARYQUESTIONS

Ravina looks in a mirror and sees the top of a building. His eyes are 1.25 m above ground level, as shown in the following diagram.

If Ravina is 1.5 m from the mirror and 181.5 m from the base of the building, how high is the building?

Solution to problem So, the height of the building is 150 m.QUESTION

ANSWER

P= 7.2

Q= 6.4QUESTION

SOLUTION

Two similar spheres made of the same material have weights of 32kg and 108 kg respectively. If the radius of the larger sphere is 9cm, find the radius of the smaller sphere.

QUESTIONSOLUTIONWe may take the ratio of weights to be the same as the ratio of volumes.

Ratio of volumes (k3) = =

Ratio of corresponding lengths (k) = = Radius of smaller sphere = 9 = 6cm

26Bibliography The information in this presentation were taken from the projects submitted by our class on the topic of similarity.

BY: Suhail Lalji Fatema Sharrif Masoomali Fatehkia Ravina Pattni Mohammed Jaffer

WE HOPE THAT THE PRESENTATION WAS GOOD AND INFORMATIVE.

THANK YOU