ESSENTIAL GEOMETRYSIMILARITY AND CONGRUENCY

HEMANGI MODI

10 B

HEMANGI MODI10BMATH PROJECT

SUB-TOPICS OVERLOOKED:

Similar Figures……………………………………………………….. Page 3

Daily life examples…………………………………………………… Page 5

Similar triangles……………………………………………………… Page 7

Perimeters and areas of similar triangles…………………………….. Page 10

Congruent figures……………………………………………………. Page 13

3-D Shapes…………………………………………………………… Page 15

Summary……………………………………………………………… Page 17

Answers……………………………………………………………….. Page 18

HEMANGI MODI10BMATH PROJECT

SIMILARITY AND CONGRUENCY:Similarity and congruency describe relationship between shapes or objects.

Similar figures:

These are figures that have the same shape but are of two different sizes. Their corresponding sides are proportional. They may have different positions and orientations.

Examples: The two pairs of figures below are similar.

Another good example;

4 cm

2 cm

B

A 4 cm 14 cm 8 cm

7 cm

225° 225°

45° 45°

5 cm 10 cm

B is an enlargement of A. The lengths have doubled, but the angles have stayed the same.

HEMANGI MODI10BMATH PROJECT

Remember: For any pair of similar figures, corresponding sides are in the same ratio and corresponding angles are equal.

Examples:

1.) In the pair of similar figures below,

And

Therefore i.e., the sides are in the same ratio.

These facts can be used when solving problems.

2.)

wxyz and WXYZ are similar figures. What is the length of XY?

The answer is 4.5cm. Here's how to work it out. Because the shapes are similar we can write

xy = (4 × 9) ÷ 8 = 4.5

What is the size of angle ?

Remember that the angles in similar figures stay the same. So is 57°.

HEMANGI MODI10BMATH PROJECT

DAILY LIFE EXAMPLES:

Examples of squares and rectangles appear in most buildings, as it is the most common shape in architecture. Why is this? Because these rectangles, each with its 90 degree angles, fit together perfectly! Doors fit into their frames, windows fit into the walls, and one wing of a building fits up against another wing of the building if all are rectangles.

Another good example would be a CD, that is a 3-D circle and the case would be a rectangular prism.

The Pyramids, in Indianapolis. The pyramids are made up of pyramids, of course, and squares. There are also many 3D geometric shapes in these pyramids. The building itself is made up of a pyramid, the windows a made up of tinted squares, and the borders of the outside walls and windows are made up of 3D geometric shapes.

Almost every construction uses geometry; similarity. All blocks and bricks are made in such a way, that they fit together. This is similarity.

A Chevrolet SSR Roadster Pickup. This car is built with geometry. The wheels and lights are circles, the doors are rectangular prisms, the main area for a person to drive and sit in it a half a sphere with the sides chopped off which makes it 1/4 of a sphere.

Some figures are always similar; circles, equilateral triangles and squares are always similar. This is because they all have fixed angles and same proportions. Rectangles are sometimes similar, but not always. That is because the sides are not always proportional.

SP

RQ

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q r

HEMANGI MODI10BMATH PROJECTQuestions:

1.) The rectangles pqrs and PQRS are similar. What is the length of PS?

4 cm 7 cm

2.) Which of the following statements is false?

Congruent shapes are always the same size Similar shapes are always the same size Angles in similar shapes are always the same Angles in congruent shapes are always the same

3.) Which of the following statements is true?

All circles are congruent All circles are similar Some pairs of circles are neither congruent nor similar

4.) Let triangles HJK, LMN be similar, and let HJ = 2 in, JK = 5 in, HK = 4 in, and LM = 7 in. How long are MN and LN?

?

9 cm

HEMANGI MODI10BMATH PROJECT

SIMILAR TRIANGLES:

Although similar geometric figures can be of any shape, similar triangles are the figures most often used in applications. Similar triangles are special type of similar figures because they have many theorems.

Here are four useful facts that allow us to be certain that two triangles are similar:

If the measures of the three angles of one triangle are equal to the measures of the three angles of another triangle, then those two triangles are similar.

108°

45°

For any polygons except triangles, mere equality of angles does not insure similarity. For example, the figure shows two trapezoids that have the same collection of angles. These trapezoids are not similar, however, because their dimensions are not proportional.

15 9

If two parallel lines are each intersected by a pair of non-parallel transversals (where the transversals intersect somewhere other than on one of the parallel lines), then two similar triangles are formed.

n These include alternate interior angles which are

zy x

zyx

m

16

4410

12

6

108°45°

27°

27°

Equal.

HEMANGI MODI10BMATH PROJECT

A line that is parallel to a side of a triangle, and which intersects the other sides of the triangle, will form a smaller triangle that is similar to the larger triangle.

In the example shown, side DE is parallel to side AC. Therefore, triangle DBE is similar to triangle ABC, with side DE corresponding to side AC, side DB corresponding to side AB, and side BE corresponding to side BC.

If the lengths of the sides of two triangles are proportional and the angle between the sides is equal, then the triangles are similar.

Refer to page 4

Similarity of two triangles is assured if only two pairs of angles have equal measures, but to assure similarity based on proportional sides, all three pairs of sides must be proportional. Furthermore, this test only works for triangles. The figure shows two quadrilaterals that have proportional sides but these quadrilaterals are not similar. A triangle is the only rigid polygon. Thus similar triangles are very special.

3

4

5

2

5

2

3

4

B

D E

A q

C q

HEMANGI MODI10BMATH PROJECTAs the examples below show, knowledge of some dimensions of similar triangles may allow us to determine other dimensions which in turn allow us to measure the perimeter and area of the figure.

1. Two triangles each have angles of measure 40°, 60° and 80°. The longest side of one triangle has length 6 feet, while the longest side of the other triangle has length 8 feet. What is the ratio of the perimeter of the smaller triangle (Ps) to the perimeter of the larger triangle (PL)?

The fact that the angles of one triangle have the same measures as the angles of the other triangle means that the triangles must be similar. Since the perimeter of a polygon is a dimension, then the ratio of the perimeters will be equal to the ratio of any other corresponding dimensions. The longest side of the smaller triangle corresponds to the longest side of the larger triangle. Therefore,

PsPL =

longest side of smaller trianglelongest side of larger trianlge =

6 ft8 ft

PsPL =

34

The perimeter of the smaller triangle, therefore, will equal three-fourths of the perimeter of the larger triangle.

Thus, sides of the similar triangle and the perimeter and areas are related. They have some connection with the sides.

In the next page, the relationship is shown.

Perimeter=?

Area=?

HEMANGI MODI10BMATH PROJECT

Similar Triangles: Perimeters and Areas

When two triangles are similar, the reduced ratio of any two corresponding sides is called the scale factor of the similar triangles. In the figure below, Δ ABC∼ Δ DEF.

Similar triangles whose scale factor is 2 : 1

The ratios of corresponding sides are 6/3, 8/4, 10/5. These all reduce to 2/1. It is then said that the scale factor of these two similar triangles is 2: 1.

The perimeter of Δ ABC is 24 inches, and the perimeter of Δ DEF is 12 inches. When you compare the ratios of the perimeters of these similar triangles, you also get 2 : 1. This leads to the following theorem.

Theorem 60: If two similar triangles have a scale factor of a : b, then the ratio of their perimeters is a : b.

Example 1: In the Figure below, Δ ABC∼ Δ DEF. Find the perimeter of Δ DEF

The Figure shows two similar right triangles whose scale factor is 2: 3. Because GH ⊥ GI and JK ⊥ JL, they can be considered base and height for each triangle. You can now find the area of each triangle.

HEMANGI MODI10BMATH PROJECT

Now you can compare the ratio of the areas of these similar triangles.

This leads to the following theorem:

Theorem 61: If two similar triangles have a scale factor of a : b, then the ratio of their areas is a2 : b2.

Example 2: In Figure below, Δ PQR∼ Δ STU. Find the area of Δ STU.

.

Questions:

Area GHI= 12×6×8= 24 inches2

Area JKL= 12×9×12=54 inches2

HEMANGI MODI10BMATH PROJECT

Questions:

1.) The perimeters of two similar triangles are in the ratio 3: 4. The sum of their areas is 75 cm2. Find the area of each triangle.

2.) The areas of two similar triangles are 45 cm2 and 80 cm2. The sum of their perimeters is 35 cm. Find the perimeter of each triangle.

3.) The ratio of the corresponding sides of two similar triangles is 5 : 11. What is the ratio of

their areas?

1:1 5 : 11 121 : 25 25 : 121

4.) The perimeters of two similar triangles are 36 cm and 63 cm respectively. What is the ratio of their areas?

49 : 16 4 : 7 16 : 49 7 : 4

5.) The perimeters of two similar triangles ABC and DEF are 9 cm and 6 cm respectively. If BC = 3 cm, then find the measure of EF

4 cm 2 cm 12 cm 7 cm

HEMANGI MODI10BMATH PROJECT

Congruent figures:

Two figures are congruent when they are exactly the same size and shape. If the corresponding sides and angles are congruent, then the figures themselves are congruent.

The two figures are congruent as the corresponding parts

have equal measures and they are the same shape and are of

same size.

Often the solving of some geometry problem is possible only if it is known that two triangles are congruent. If each of the six parts of a triangle (the three sides and the three angles) can be shown to have the same measures as the six parts of another triangle, then those triangles must be congruent. We seldom know the measures of all six parts of a triangle, but fortunately it sometimes is possible to show that two triangles are congruent by knowing that only three parts of one triangle have the same measures as their corresponding parts in the other triangle.

The four cases (theorems), where the equality of the measures of the three parts implies that the two triangles are congruent are the following:

1.) Three sides. (This case is identified by the abbreviation SSS)

2.) Two sides and the included angle. (This case is identified by the abbreviation SAS). Two sides of a triangle are said to “include” an angle if each of those sides is a side of the angle. Example;

Sides b and c include angle A

b

c

3.) Two angles and the included side. (This case is identified by the abbreviation ASA). Two angles of a triangle are said to “include” a side of the triangle if the side is a side of each of the angles. Example;

Angles A and C include side b.

b

A

B

C

C

B

A

k a g

HEMANGI MODI10BMATH PROJECT

4.) The right angle-hypotenuse-side (RHS) principle. Two right-angled triangles are congruent if the hypotenuses and one pair of corresponding sides are equal.

When attempting to prove that two triangles are congruent, one must be sure that the parts of the two triangles that are being shown to be equal fall under the SSS case, or the SAS case, or the ASA case. It is possible for two triangles to have three pairs of equal parts and not be congruent. For example;

Although the corresponding sides are equal, the two triangles are not congruent because the sides do not include the angle.

d

Question:

1.) For each of the following pairs of triangles, state whether they are congruent. If they are, give a reason for your answer (SSS, SAS, AAS or RHS).

Pair 1

Pair 2

f

A

HEMANGI MODI10BMATH PROJECT

3-D FIGURES:

The result of perimeters and areas of similar figures also applies to the surface areas of three-dimensional objects. A 3-D figure mostly has at least a triangle, circle, or rectangle-all similar figures.

Thus, the conditions applied are the same; all the corresponding angles should be equal and all corresponding sides should be proportional.

Volumes of similar objects:

When solid objects are similar, one is an accurate (exact) enlargement of the other.

If two objects are similar and the ratio of corresponding sides is k, then the ratio of their volume isk3.

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

Example:

6 cm

3 cm

Two similar cylinders have heights of 3cm and 6cm respectively. If the volume of the small cylinder is 30cm3, find the volume of the larger cylinder.

If scale factor=k, then ratio of heights (k) =63=2

30 cm3

HEMANGI MODI10BMATH PROJECT∴ ratio of volumes (k3)= 23

= 8

And volume of larger cylinder= 8×30

= 240 cm3

This is how the sides of a figure and its volume are related

Questions:

1.) Two similar jugs have heights of 4cm and 6cm respectively. If the capacity of the smaller jug is 50cm3, find the capacity of the larger jug.

2.) Find the length marked by the letter x:

X

3.) The area of the bases of two similar glasses are in the ratio 4:25. Find the ratio of their volumes.

4.) A container has a surface area of 5000cm2 and a capacity of 12.8 litres. Find the surface area of a similar container which has a capacity of 5.4 litres.

5.) A cylindrical can has a circumference of 40cm and a capacity of 4.8 litres. Find the capacity of a similar cylinder of circumference 50cm.

6.) In the figure below, the objects are similar and the number written inside represents the volume of the object incm3. The numbers on the outside give linear dimensions in cm. find the unknown volume, V.

5 20 15 v

24

HEMANGI MODI10BMATH PROJECT

SUMMARY:

Two geometric figures are congruent if they have exactly the same shape and size.

Two geometric figures are similar if they have exactly the same shape.

Congruent geometric figures are also similar.

The difference between congruence and similarity is that, two shapes are similar if one is congruent to an enlargement of the other, whereas congruent shapes must have identical lengths

Whether shapes are congruent or similar can be found by comparing their internal angles and the length of their sides.

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

If two figures are similar and the ratio of corresponding sides is k, then the ratio of their areas is k 2.

If two objects are similar and the ratio of corresponding sides is k, then the ratio of their volumes is k3.

HEMANGI MODI10BMATH PROJECT

ANSWERS: Similar figures- page.6

1.) 15.75 cm2.) The second one is false. Similar shapes are not always the same size.3.) The second statement is true. All circles are always similar.4.) LN= 14 inches

MN=17.5 inches

Perimeter and area of similar triangles-page.12

1.) Area 1=27 cm2

Area 2=48 cm2

2.) Perimeter 1= 15 cmPerimeter 2= 20 cm

3.) The correct answer is 25: 121, the last option.4.) The correct answer is 16: 49, the third option.5.) The correct answer is 2cm, the second option

3-D figures, page 17

1.) 16834

cm3

2.)4 cm3.)8: 1254.)281212

cm2

5.)938 litres6.)540 cm3

GRADE X - ASSESSMENT - SIMILARITY AND CONGRUENCY

Writing skill

Drawing skill

Formatting skill Polygon Triangle

3D shapes Questions

Examples Total

HEMANGI MODI10BMATH PROJECT

3 2 3 2 2 3 3 2 20creativity

flow neatness

pictures labeling

page paragraphs

math equations

always-similar

polygons appln

theorems

applns

perimeter vol, SA

3D shapes

congruency variety

ex+ans

3 2 2 2 2 2 2 1 16