Exploring Geometrical Figures

Congruent Objects have the same size, same shape and same angle. If you place a congruent object in front of a mirror, the image that you see is congruent or "equal" to the object. They completely overlap each other.

Any five pairs of congruent objects you use daily are:

a) Biscuits of same pack

b) All A4 sheets

c) Two bangles

d) Two balls of same brand and same size

e) Two earrings of the same set

We know that two figures are “congruent” if they have the same size, same shape and same angle.

When two figures are “similar”, their angles are same. But that does not mean they have to be congruent. Whereas “Congruent” figures are “always similar”.

Example 1:

Above two figures are congruent as they have same size, same shape and same angle and all congruent figures are similar.

Example 2:

Figures given above are similar as they have same angles but are not congruent.

Two figures shown above are similar to each other as they have same angles.

On rotating figure on left by 180, we get:

We see that the rotated figure is similar to figure on the right.

Given: Δ ABC ≅ Δ NMO

Congruent sides:

a) AB = NM

b) BC = MO

c) AC = NO

Congruent angles:

a) ∠A = ∠N

b) ∠B = ∠M

c) ∠C = ∠O

True

On rotating the square by 45, the length of side, angle and shape remains the same. Hence, they are congruent.

True

For a right angled triangle to have hypotenuse equal to 5 cm, the other two sides have to be 3 cm and 4 cm as the only possible way. As, by Pythagoras theorem for Right angled triangles. Thus, any two right triangles with hypotenuse 5cm have same angles and are congruent.

True

Any two circles of radii 4 cm would completely overlap each other when put front of mirror due to same radii of 4 cm. Hence they are congruent.

False

Two equilateral triangles of side 4 cm and angle 60 each but labeled as ΔABC and ΔLHN would still be congruent as they have same sides and same angles.

True

Mirror image of a polygon would have same sides and angles as the original and thus, is congruent to the original.

Perimeter = 2(l + b)

= 2(2 + 1)

= 2(3)

= 6 cm

Area = l×b

= 2×1

=

A figure similar to above figure is shown below:

Perimeter =

=

=

= 12 cm

Area =

=

=

Ratio of the sides of the two figures (fig 2:fig 1) = 2:1

Ratio of the areas of the two figures (fig 2:fig 1) = 3:1

Ratio of the perimeter of the two figures (fig 2:fig 1) = 2:1

Given, Distance between every two pillars = 1 m

Height of the last pillar = 10.5 m

Number of pillars used to hold a slant iron ladder = 7

Distance between the tip of ladder on floor to the last pillar = 7 × 1 = 7 cm

Height of last pillar = 10.5 cm

Formula Used:

Pythagoras Theorem:

⇒

Length of slant ladder =

=

=

= 12.619 cm (ans)

Let the height of the building be x cm.

In ABC,

In ADE,

⇒

⇒

⇒

Height of building = 9 cm.

Given, Dilation of scale factor = 3

Let’s draw a square with each side = 3 cm

Now on dilating the square using scale factor = 3, then

Each side of the square becomes = cm

We know that two figures are similar if their angles are same.

In the two squares drawn above we know that each angle = 90

Hence, both the figures are similar.

(i) 9; F,G,J,L,N,P,Q,R,S

(ii) 13; A,B,C,D,E,K,M,T,U,V,W,Y,Z

(iii)3; H,I,X

(iv) 1;O

(v) 1;O

When a shape it looks the same after some rotation by a partial turn, then its said to have rotational symmetry

(vi) 5; H,J,N,O,X

A figure has point symmetry when it looks the same when up-side-down, or when rotated by 180.

Line of symmetry = 2

Line of symmetry = 3

Line of symmetry = 8

Line of symmetry = 10

Line of symmetry = 12

Line of symmetry = 14

Line of symmetry = 16

All of the above figures have point symmetry.

Some natural objects with faces which have atleast one line of symmetry are:

a) Moon

b) Sun

c) Human body

d) Star Fish

e) Plant Leaf

A tessellation is created when a shape is repeated over and over again to cover the plane without any overlaps or gaps.

Basic figure used-Rectangle

Basic figure used-Triangle

Basic figure used-Square