Skip to content Skip to navigation

Connexions

You are here: Home » Content » Similarity

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

In these lenses

  • GETSenPhaseMaths display tagshide tags

    This module is included inLens: Siyavula: Mathematics (Gr. 7-9)
    By: SiyavulaAs a part of collection: "Mathematics Grade 9"

    Collection Review Status: In Review

    Click the "GETSenPhaseMaths" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Similarity

Module by: Siyavula Uploaders. E-mail the author

MATHEMATICS

Grade 9

ALGEBRA AND GEOMETRY

Module 10

SIMILARITY

ACTIVITY 1:

To practically investigate the conditions for similarity

1. The pentagons ABDEF and LCMRK are given (A-6). LCMRK is an enlargement of ABDEF. What is the scale factor by which ABDEF were enlarged to give LCMRK?

2. Write down the ratios between the corresponding pairs of sides of ABDEF and LCMRK.

3. Write down the relationship between the corresponding pairs of angles of the two figures.

4. These two figures are not congruent. What do we call them?

5. Name as many as possible examples of this phenomenon in real life.

Similar figures:

  • The pentagons in the activity above are similar. They have the same form, but do not have the same size.
  • Their corresponding angles have the same magnitudes.
  • Their corresponding sides are in the same ratio.

Therefore LKAF=KRFE=MRDE=CMBD=CLBA=31LKAF=KRFE=MRDE=CMBD=CLBA=31 size 12{ { { ital "LK"} over { ital "AF"} } = { { ital "KR"} over { ital "FE"} } = { { ital "MR"} over { ital "DE"} } = { { ital "CM"} over { ital "BD"} } = { { ital "CL"} over { ital "BA"} } = { {3} over {1} } } {} This constant ratio also is the scale factor of the enlargement.

  • We say that ABDEF  LCMRK. Note that the order of the letters is in the same order of the angles which are equal and the sides which are in proportion. (The symbol for similarity is )

Homework assignment

1. Measure the lengths of the sides and the magnitudes of the angles in the following figures (A-7) and decide whether they are similar or not. If the two figures are not similar, give a reason why they are not similar.

2. If the corresponding angles of two quadrilaterals are equal, are they necessarily also similar?

3. If corresponding sides of two quadrilaterals are proportional, are they necessarily also similar?

  • In the homework assignment above you saw that, for quadrilaterals to be similar, both conditions of similarity must be satisfied. In other words, the corresponding angles must be equal and the corresponding sides must be proportional. Do the same conditions also apply to triangles?

ACTIVITY 2:

To practically investigate the conditions for similarity in triangles

[LO 3.5]

Figure 1
Figure 1 (graphics1.png)

Construct ΔABC and ΔDEF. Calculate the magnitudes A and E.

1.2 Are the corresponding angles of the two triangles equal?

1.3 Complete the following:

ABED=ABED= size 12{ { { ital "AB"} over { ital "ED"} } ={}} {}....................

BCDF=BCDF= size 12{ { { ital "BC"} over { ital "DF"} } ={}} {}....................

ACEF=ACEF= size 12{ { { ital "AC"} over { ital "EF"} } ={}} {}....................

1.4 Are the corresponding sides of the two triangles proportional?

1.5 Are the two triangles similar?

1.6 Complete the following: If the corresponding angles of two triangles are equal, their corresponding sides are necessarily also always ......................... This means that, if the corresponding angles of triangles are equal the triangles are .........................

2.1 Construct the following two triangles:

Figure 2
Figure 2 (graphics2.png)

2.2 Are the sides of the two triangles proportional?

2.3 Measure all the angles of ΔABC and ΔMOR. What do you find?

2.4 Is ΔABC  ΔMOR?

2.5 Complete the following: If the corresponding sides of two triangles are proportional then their corresponding ..................................... are equal. That therefore means that, if the corresponding sides of two triangles are proportional, the triangles are.....................................

  • We therefore see that with triangles only one of the conditions of similarity have to be present for triangles to be similar.
  • That means that, if the three angles of one triangle are equal to the three angles of the other triangle, then the corresponding sides of the two triangles are proportional and the triangles are therefore also similar.
  • It also means that, if the corresponding sides of the triangles are proportional, then the corresponding angles of the two triangles are equal and the triangles are therefore also similar.

Homework assignment

1. The following pairs of triangles are given. State whether they are similar or not and give reasons for you answer. If the two triangles are similar, calculate the lengths of the sides not given and also the magnitudes of the angles not given in the figure.

Example:

Figure 3
Figure 3 (graphics3.png)

C = F = 60°Δ ABC  ΔEDF ()

AB = 4 cmAC = 5 cm (pyth)EF = 10 cm

1.1.

Figure 4
Figure 4 (graphics4.png)

1.2

Figure 5
Figure 5 (graphics5.png)

1.3.

Figure 6
Figure 6 (graphics6.png)

1.4

Figure 7
Figure 7 (graphics7.png)

2.

Figure 8
Figure 8 (graphics8.png)

2.1 Complete the following:

In ΔAOB and ΔDOE:

Reason

.......... = .......... (...........................................................................)

.......... = .......... (...........................................................................)

Δ.............. .............. ( )

2.2 Now calculate the lengths of the sides not given in the figure.

3.

Figure 9
Figure 9 (graphics9.png)

3.1 Complete the following:

In Δ.......... and Δ..........

.......... = .......... (...........................................................................)

.......... = .......... (...........................................................................)

Δ.............. .............. ( )

3.2 Calculate the lengths of the following:

3.2.1 HE

3.2.2 EG

3.2.3 FJ

4. The height of a high vertical object can be determined by measuring the length of the shadow of a stick of known length and the shadow of the object. The following figures give the measurements which were made.

Figure 10
Figure 10 (graphics10.png)

Determine the length of the flagpole.

ASSESSMENT TASK:

To use similarity to calculate the height of an object:

  • Work together in pairs.

1. The following are needed:

  • A measuring tape of at least 5 m
  • A mirror
  • A ruler
  • A Koki pen

2. You do the following:

  • Look for two high vertical objects on the school grounds; for example a netball pole, a lamp pole, rugby poles or a flagpole. Look for objects of which the heights are normally difficult to measure using normal methods.
  • Draw two thin lines on the mirror using the Koki pen so that the lines are perpendicular to each other.
  • Place the mirror a distance from the object on level ground.
  • One person should then step back and look in the mirror and change his / her position until the top of the object is precisely on the point of intersection of the two lines in the mirror.

3. Measure the following:

  • the height of the eyes above the ground of the person who looked in the mirror;
  • the distance between the person who looked in the mirror and the point of intersection of the lines in the mirror’
  • the distance between the object and the point of intersection of the lines in the mirror.

Results:

1. Copy the table on folio paper and complete it:

Table 1
The object of which the height is measured The height of the eyes of the person above the ground. Distance between the person and the point of intersection of the lines in the mirror Distance between the point of intersection of the lines in the mirror and the object Calculate the height of the object correct to the nearest cm
         
         

2.

PKSLV

In the sketch PK is the height of the eyes of the person, S is the position of the mirror and VL is the height of the object, which is measured. Explain why ΔPKS  ΔVLS.

3. In this task the measurements can be inaccurate. Explain which mistakes could have been made, which could influence the accuracy of the height of the object measured.

Assessment

Table 2
LO 3
Space and Shape (Geometry)The learner will be able to describe and represent characteristics and relationships between two-dimensional shapes and three-dimensional objects in a variety of orientations and positions.
We know this when the learner :
3.1 recognises, visualises and names geometric figures and solids in natural and cultural forms and geometric settings, including:3.1.1 regular and irregular polygons and polyhedra;3.1.2 spheres;3.1.3 cylinders;3.2 in contexts that include those that may be used to build awareness of social, cultural and environmental issues, describes the interrelationships of the properties of geometric figures and solids with justification, including:3.2.1 congruence and straight line geometry;3.3 uses geometry of straight lines and triangles to solve problems and to justify relationships in geometric figures;3.4 draws and/or constructs geometric figures and makes models of solids in order to investigate and compare their properties and model situations in the environment;
3.5 uses transformations, congruence and similarity to investigate, describe and justify (alone and/or as a member of a group or team) properties of geometric figures and solids, including tests for similarity and congruence of triangles.

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks