Chapter 12: Practical Geometry Symmetry and Visualising Solid Shapes

Q101Written answer

Construct an equilateral triangle ${\text{ABC}}$ of side ${\text{6}}\,{\text{cm}}$.

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Worked solution

Ans: Steps of construction:
(1) Draw a line segment ${\text{AB = 6}}\,{\text{cm}}$

(2) Draw an arc of radius ${\text{6}}\,{\text{cm}}$ from point ${\text{A}}.$
(3) Now, draw another arc of radius ${\text{6}}\,{\text{cm}}$ from point ${\text{B}}$ to cut previous arc at ${\text{C}}.$
(4) Join ${\text{A}}$ to ${\text{C}}$ and ${\text{B}}$ to ${\text{C}}$.

Hence, $\Delta {\text{ABC}}$ is the required triangle of side ${\text{6}}\,{\text{cm}}$.

Q102Written answer

By what minimum angle does a regular hexagon rotate so as to coincide with its original position for the first time?

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Ans:

A regular hexagon must be rotated through a minimum angle of ${60}^{\circ} $. So, that it can coincide with its original position for the first time. Because the angle of rotation of hexagon is,

$\dfrac{{{{360}^\circ }}}{{{\text{ Number of sides }}}} = \dfrac{{{{360}^\circ }}}{6} = {60^\circ }$

Q103Written answer

In each of the following figures, write the number of lines of symmetry and order of rotational symmetry.

{Hint: Consider these as 2-D figures not as 3-D objects.}

(a)

(b)

(c)

(d)

(i)

(v)

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Worked solution

Ans: The number of lines of symmetry is $1$.

The order of rotational symmetry is $1$.

Ans: The number of lines of symmetry is $1$.

The order of rotational symmetry is $1$.

Ans: The number of lines of symmetry is $1$.

The order of rotational symmetry is $1$.

Ans: The number of lines of symmetry is $2$.

The order of rotational symmetry is $2$.

(e)

Ans: The number of lines of symmetry is $1$.

The order of rotational symmetry is $1$.

(f)

Ans: The number of lines of symmetry is $0$.

The order of rotational symmetry is $1$.

(g)

Ans: The number of lines of symmetry is $0$.

The order of rotational symmetry is $1$.

(h)

Ans: The number of lines of symmetry is $0$.

The order of rotational symmetry is $3$.

Ans: The number of lines of symmetry is $4$.

The order of rotational symmetry is $4$.

(j)

Ans: The number of lines of symmetry is $1$.

The order of rotational symmetry is $1$.

(k)

Ans: The number of lines of symmetry is $0$.

The order of rotational symmetry is $1$.

(l)

Ans: The number of lines of symmetry is $0$.

The order of rotational symmetry is $1$.

(m)

Ans: The number of lines of symmetry is $0$.

The order of rotational symmetry is $2$.

(n)

Ans: The number of lines of symmetry is $0$.

The order of rotational symmetry is $1$.

(o)

Ans:The number of lines of symmetry is $1$.

The order of rotational symmetry is $1$.

(p)

Ans: The number of lines of symmetry is $0$.

The order of rotational symmetry is $1$.

(q)

Ans: The number of lines of symmetry is $5$.

The order of rotational symmetry is $5$.

(r)

Ans: The number of lines of symmetry is $0$.

The order of rotational symmetry is $1$.

(s)

Ans: The number of lines of symmetry is $3$.

The order of rotational symmetry is $3$.

Ans: The number of lines of symmetry is $1$.

The order of rotational symmetry is $1$.

(u)

Ans: The number of lines of symmetry is $0$.

The order of rotational symmetry is $10$.

Ans: The number of lines of symmetry is $3$.

The order of rotational symmetry is $3$.

(w)

Ans: The number of lines of symmetry is $0$.

The order of rotational symmetry is $1$.

Q104Written answer

In the figure 12.24 of a cube,

(i) Which edge is the intersection of faces ${\mathbf{{\rm E}FG{\rm H}}}$ and ${\mathbf{EFBA}}$?

(ii) Which faces intersect at the edge ${\mathbf{FB}}$?

(iii) Which three faces form the vertex ${\mathbf{A}}$?

(iv) Which vertex is formed by the faces ${\mathbf{ABCD,ADHE}}$ and ${\mathbf{CDHG?}}$

(v) Give all the edges that are parallel to edge ${\mathbf{AB}}$.

vi) Give the edges that are neither parallel nor perpendicular to edge ${\mathbf{BC}}$.

(vii) Give all the edges that are perpendicular to the edge ${\mathbf{AB}}$.

(viii) Give four vertices that do not all lie in one plane.

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Worked solution

Ans: From the given figure, we can observe that ${\text{EF}}$ is the intersection of faces ${\text{EFGH}}$ and ${\text{EFBA}}{\text{.}}$

Ans: From the given figure, we can observe that faces ${\text{EFBA}}$ and ${\text{FBCG}}$ intersect at edge ${\text{FB}}{\text{.}}$

Ans:Faces ${\text{ABFE}},{\text{ADHE}}$ and ${\text{ABCD}}$ form the vertex ${\text{A}}$.

Ans: Vertex ${\text{D}}$ is formed by the faces ${\text{ABCD}}$, ${\text{CDHG}}$ and ${\text{ADHE}}$.

Ans: The edges parallel to edge ${\text{AB}}$ are ${\text{CD}},{\text{EF}}$ and ${\text{HG}}$.

Ans: From the given figure, we can observe that edges ${\text{AE}},{\text{EF}},{\text{GH}}$ and ${\text{HD}}$ are neither parallel nor perpendicular to edge ${\text{BC}}$.

Ans: From the given figure, we can observe that edges ${\text{AE}},{\text{BF}},{\text{AD}}$ and ${\text{BC}}$ are perpendicular to edge ${\text{AB}}$.

Ans: Vertices ${\text{A}},{\text{B}},{\text{G}}$ and ${\text{H}}$ do not lie in one plane.

Q105Written answer

Draw a net of a cuboid having same breadth and height, but length double the breadth.

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Worked solution

Ans: Let $x$ be the breadth and height.

Then, length is $2x.$

Required net of a cuboid will be,