Practical Geometry Symmetry And Visualising Solid Shapes

The basic condition for the formation of triangle is that the sum of two sides should be greater than the third side.

Now let us take each option separately,

For option A:

1.8 + 2.6 = 4.4 which is the third side.

For option B:

2 + 3 = 5 which is greater than the third side.

For Option C:

2.4 + 2.4 = 4.8 which is very less than the third side.

For option D:

3.2 + 2.3 = 5.5 which is equal to the third side.

So clearly it is seen that only option B is satisfying the condition.

So the correct answer is B.

For a proper triangle the sum of all interior or internal angles should be equal to 180°.

Now let us take each option separately,

For option A:

110° + 40° = 140° and so the third angle is 40°.

For Option B:

70° + 115° = 185° which is greater than 180° which is not correct.

For option C:

135° + 45° = 180°. Here the sum is already 180° without considering the third angle which is incorrect.

For option D:

90° + 90° = 180°. Here the sum is already 180° without considering the third angle which is incorrect.

So it is clearly observed that only option A is satisfying the angle condition of the triangle.

So the correct answer is A.

Line of symmetry is the line which divides a figure into two equal halves which are mirror image of each other.

So for the given figure, it has six grooves or inside edge through which we can draw three lines of symmetry.

The figure also has six outside edge through which we can draw three lines of symmetry.

So total six lines of symmetry for the figure is six.

The figure is as follows:

So the correct answer is C.

Line of symmetry is the line which divides a figure into two equal halves which are mirror image of each other.

For this figure has six sides and so total three lines of symmetry can be drawn over this figure.

The figure is as follows:

So the correct answer is B.

The number of times a figure fits into itself in one full turn is called as order of rotational symmetry.

So for the given figure the order of rotational symmetry is six.

So the correct answer is C.

The number of times a figure fits into itself in one full turn is called as order of rotational symmetry.

So for the given figure the order of rotational symmetry is two

So the correct answer is B.

From the figure shown we can conclude that:

The base of the figure is a rectangle.

The sides of the figure are triangles.

So the given figure is a combination of rectangle and triangle.

So the figure is a rectangular pyramid.

So the correct answer is B.

Prism is a combination of triangular faces and parallelogram faces.

So in the given figure there are two triangular faces and parallelogram faces.

So this figure is a combination of triangle and prism.

So the correct answer is C.

The faces of pyramid can be triangular or rectangular. But we have to choose only one correct option.

So the correct answer is D.

From the figure of a cone it is seen that it has only one vertex.

So the correct answer is C.

A 3-D figure is defined as the figure which can be specified in all three dimensional axis that is x-axis, y-axis and z-axis.

A 2-D figure can be represented only on a plane.

From the available option Sphere is the only figure can be visualized in space.

All the remaining figures in the option are 2-D figures.

So the correct answer is B.

The total number of edges of a cylinder are two.

The figure of cylinder is as follows:

So the correct answer is C.

From the given options prism is the only figure which has two opposite identical faces and other faces as parallelogram.

For a pyramid the faces are triangular.

For cone it is curved surface.

Sphere is a whole figure.

So the correct answer is A.

From the given options cone is the only figure which has a circular base, one curved surface and one vertex.

Cylinder and sphere does not have a vertex.

Prism does not have only one vertex.

So the correct answer is A.

When three cubes are placed end to end then the dimensions of the resulting solid would be more than the cube.

The new cuboid will be having the following dimensions:

12 cm × 4 cm × 4 cm.

So the correct answer is A.

If we cut a corner of a cube as shown in the figure, then we will get a cut piece in the form of a triangular pyramid.

So the correct answer is C.

When a right angled triangle of height 5 cm and base 3 cm is rotated about its height in full turn, then we will get a cone of height 5 cm and base of radius 3 cm.

So the correct answer is A.

When a right angled triangle of height 5 cm and base 3 cm is rotated about its base in full turn, then we will get a cone of height 3 cm and base of radius 5 cm.

So the correct answer is D.

When a torch is pointed towards one of the vertical edges of a cube, you get a shadow of cube in the shape of a rectangle.

Hence the correct answer is B.

For a right angled triangle, minimum condition to be satisfied is the Pythagoras Theorem which is stated mathematically as below:

(Hypotenuse) ² = (Base) ² + (Perpendicular) ²

So let us try each and every option,

For option A:

26² = 9² + 16²

676 = 81 + 256

676 ≠337

So this option is not correct.

For option C:

3.9² = 3.6² + 1.5²

15.21 = 12.96 + 2.25

15.21 = 15.21

So this option is correct.

For option D:

26² = 7² + 24²

676 = 49 + 574

676 ≠ 623

So this option is not correct.

So the correct answer is C.

If we consider BC to be the base of the triangle,

The sum of two sides AB and AC is not greater than the third side. Hence triangle cannot be formed.

In the second option the sum of two angles is 200° which is greater than 180° and hence triangle is not possible as sum of all interior angles of a triangle should be equal to 180°.

In the third option a unique triangle is possible.

For the fourth option, the side length of third side is variable and hence multiple triangle can be formed and hence no unique triangle can be constructed.

So the correct answer is C.

Line of symmetry is the line which divides a figure into two equal halves which are mirror image of each other.

From all the options mentioned, there exists only one figure which when cut by the line of symmetry will be divided in two equal halves.

So the correct answer is T.

Reflection means the figure should be just in opposite orientation like the image in a mirror that is the image is just flipped.

So from the given option only option A is the proper reflection example.

So the correct answer is A.

A net is 2-D figure which when folded in a particular axis gives a 3-D figure.

So for the net to form a cube, the shape of the interlinking blocks of nets should be a square which is present only in option B.

So the correct answer is B.

A net is 2-D figure which when folded in a particular axis gives a 3-D figure.

So for the net to form a cylinder, the shape of the interlinking blocks of nets should contain a rectangle which is present only in option C.

So the correct answer is C.

In the given figures,

H has only two lines of symmetry.

E has only one line of symmetry.

Z has no line of symmetry.

So the only figure left is O

O has infinite lines of symmetry.

So the correct answer is B.

As per the steps given in the question, if we open the piece of paper then we will get a figure as shown in option A.

So the correct answer is A.

From the given figures the figure which will have the top, side and front view as the triangle is the triangular pyramid.

For Cylinder the front view would be rectangle.

Similarly, it will vary for other figures.

So the correct answer is C.

An isosceles triangle has only one line of symmetry drawn from the vertex touching the base as shown in the figure:

In an isosceles right triangle, the number of lines of symmetry is one.

A rhombus has two lines of symmetry along its diagonals.

It has a rotational symmetry of the order two.

Rhombus is a figure that has two lines of symmetry and has a rotational symmetry of order two.

The triangle which has only one line of symmetry and no rotational symmetry is the isosceles triangle.

Isosceles triangle is a figure that has a line of symmetry but lacks rotational symmetry.

Scalene Triangle has all sides unequal and hence it has not lines of symmetry or rotational symmetry.

Scalene Triangle is a figure that has neither a line of symmetry nor a rotational symmetry.

The only two letters which have only one line of symmetry and those which interchange to each other when rotated through 180° are M and W.

M and W are the capital letters of English alphabets that have one line of symmetry but they interchange to each other when rotated through 180°.

The edge of a cuboid is the common portion of two adjacent faces.

The common portion of two adjacent faces of a cuboid is called edge.

A plane surface of a solid enclosed by edges is called face.

The corners of solid shapes are called its vertices.

Sphere is the only solid figure with zero vertex, zero edge and 1 curved surface.

A solid with no vertex is sphere.

A triangular prism has five faces, nine edges and six vertices.

A triangular pyramid has four faces, six edges and four vertices.

A square pyramid has five faces, eight edges and five vertices.

Out of five faces of a triangular prism, three are rectangles and two are triangles.

The base of a triangular pyramid is a triangle.

Out of five faces of a square pyramid, four are triangles and one is square.

Out of five faces of a rectangular pyramid four are triangles and base is rectangle.

Each of the letters H, N, S and Z has a rotational symmetry of order two.

Order of rotational symmetry of a rectangle is two.

Order of rotational symmetry of a circle is two.

A cuboid is a solid figure which is bounded by six rectangular faces.

Each face of a cuboid is a rectangle.

Line of symmetry for an angle is its angle bisector.

A parallelogram has no line of symmetry.

8

The number of times a figure fits into itself in one full turn is called as order of rotational symmetry.

Order of rotational symmetry of is 8.

A scalene triangle has no lines of symmetry.

As all the sides of a scalene triangle are of different lengths.

Cuboid is a rectangular prism.

Rectangular prism and cuboid are same solids.

Sphere is the only solid figure with zero vertex, zero edge and 1 curved surface.

is a net of a cone.

→ Circumference of circle = 2πr.

is a net of a triangular prism.

For the given triangle, two sides are equal which means it is an isosceles triangle.

Order of rotational symmetry of is two.

The number of cubes that are not visible are 20.

False

Though the angles are unique for the triangle, but the length of the sides can differ and hence the number of triangles that can be drawn with this angles will also be more.

So the given statement is false.

We can draw more than one triangle whose angles are 70°, 30° and 80°.

True

The given statement is true because the distance between two parallel lines is a constant value as distance between two lines decreases for two intersecting lines till point of intersection and then thereafter the distance increases.

False

A circle has infinite lines of symmetry and hence the given statement is false.

False

For an angle, only its angle bisector is its line of symmetry. So the given statement is false.

True

For regular polygon with n sides, the lines of symmetry is also n. So hexagon has six sides and hence the number of line of symmetry is also six.

So the given statement is true.

True

An isosceles trapezium has only one line of symmetry passing through the midpoints of the two parallel sides of the trapezium.

So the given statement is true.

False

A parallelogram has no lines of symmetry.

So the given statement is false.

False

The order of rotational symmetry for a rhombus is two.

So the given statement is false.

False

An equilateral triangle has 3 lines of symmetry.

So the given statement is false.

False

Order of rotational symmetry of a semi circle is one.

So the given statement is false.

False

In oblique sketch of the solid, the measurements are not kept proportional.

So the given statement is false.

False

An isometric sketch always has a proportional length.

So the given statement is false.

True.

A cylinder has 3 faces - 2 circle ones and a rectangle. It has 2 edges and no vertices (no corners). Hence, the given statement is true.

The given statement is true.

A square pyramid has four triangular faces and one square base. It is also shown in the image:

A pyramid has at least four vertices for triangular pyramid.

So the given statement is false.

The figure is shown below:

The given statement is true.

The figure below shows a triangular prism.

The name of a pyramid depends upon the base of the pyramid.

The given statement is true. The figure below shows the different pyramids:

A rectangular pyramid has only 1 rectangular face and 4 triangular faces.

As shown below:

So, the given statement is false.

The given statement is true.

Rectangular prism and cuboid refer to the same solid.

As shows before:

False,

A tetrahedron has 4 triangular faces and all the 4 faces are equilateral triangles, as shown in figure:

So the given statement is false.

True.

The main difference between 2D shapes including rectangles, squares, circle and triangles.

3D shapes including cubes, cuboids, etc as shown below:

False,

The main difference between 2D shapes including rectangles, squares, circle and triangles.

3D shapes including cubes, cuboids, etc as shown below:

The given statement is false.

The given statement is true.

Because these are the line segments that connect with each other to form figures as shown below:

False.

Polyhedrons are 3D figures with rectangular base, as shown below:

The given statement is false. A cone is not a polyhedron.

False.

The given statement is false. A prism has only 1 base.

As shown below:

The given statement is true.

Let us take an example of a pentagon. Pentagon has five sides, so number of lines of symmetry for pentagon = 5.

Hexagon has six sides, so number of lines of symmetry for hexagon = 6.

The given statement is false. If the order of rotational symmetry of a figure is 4 then the angle of rotation should be 90°.

The given statement is true.

The given statement is false. Mirror reflection does not always leads to symmetry.

The given statement is true.

The center of rotation is the point where the figure is rotated.

Isometric sheet divides the paper into small equilateral triangles made up of dots or lines.

So the given statement is false.

The given statement is true because circle, the square, the rectangle and the triangle are 2-D figures and all 2-D figures are plane figures.

The solid shapes are of three-dimensional. Examples of solid shapes are cone, cube etc.

So the given statement is false.

The given statement is false.

For a triangle the sum of two sides should be greater than the third side.

(i) The three views are as follows:

(ii) The three views are as follows:

The solid is as follows:

The steps are as follows:

1. Draw a line AB of side 3.2 cm

2. Construct 90° at point B and draw a straight line through it.

3. From point A cut an arc at 90° line of length 5 cm.

4. The point where arc intersect with 90° line is the third point of triangle, C.

5. Join AC to complete the triangle.

The steps are as follows:

1. Draw a line AB of side 4.5 cm

2. Construct 90° at point B and draw a straight line through it.

3. From point B cut an arc at 90° line of length 4.5 cm.

4. The point where arc intersect with 90° line is the third point of triangle, C.

5. Join AC to complete the triangle.

The steps are as follows:

1. Draw a line I and mark a point say C outside it.

2. Take a point B on line I and join BC.

3. Draw a line parallel to line I passing through C.

4. Mark a point D on line II at a distance of 2.2 cm from C.

5. Through D draw AD ∥ BC.

The steps are as follows:

1. Draw a line AB of side 3 cm

2. Construct 45° at point B and draw a straight line through it. 45° angled line can be constructed by taking same distance from B towards A [X] and towards the 90° [Y] line. Then cut the arc from the two points.

3. From point B cut an arc at 45° line of length 3 cm.

4. The point where arc intersect with 90° line is the third point of triangle, C.

5. Join AC to complete the triangle.

The steps are as follows:

1. Draw a line BC of side 7 cm

2. From B draw an arc with a radius of 4 cm.

3. From C draw an arc with a radius of 5 cm.

4. The point where the two arc intersects is the third point of triangle, A.

5. Join AC to complete the triangle.

The steps are as follows:

1. Draw a line BC of side 5.5 cm

2. Construct 120° at point B and draw a straight line through it and extend the ray Y.

3. Draw an angle of 30° at point C and extend the ray X.

4. Extend BY and CX in such manner that they intersect to get the third point A.

5. Join AC to complete the triangle.

Steps to construct 120°:

1. Use ruler and draw a Line segment OB of any convenient length.

2. Now use compass and open it to any convenient radius. And with O as center, draw an arc which cuts line segment OB at X.

3. Again use compass and opened to the same radius (as of step 2). And with X as center, draw an arc which cuts first arc at D.

4. Again use compass and opened to the same radius (as of step 2). And with D as center, draw another arc which cuts first arc at C.

5. Join OC which is 120° line.

Steps to construct 30°:

1. Use ruler and draw a Line segment OB of any convenient length.

2. Now use compass and open it to any convenient radius. And with O as center, draw an arc which cuts line segment OB at X.

3. Again use compass and opened to the same radius (as of step 2). And with X as center, draw an arc which cuts first arc at D.

4. Now keep the compass distance same and from point D draw an arc and from point X draw an arc. The two arcs will coincide to give 30° line.

5. Draw a line from O to intersection of two arcs.

The steps are as follows:

1. Draw a line AB of side 6 cm

2. From B draw an arc with a radius of 6 cm.

3. From A draw an arc with a radius of 6 cm.

4. The point where the two arc intersects is the third point of triangle, C.

5. Join AC to complete the triangle.

Angle of rotation for regular hexagon = 360° / Number of sides

= 360° / 6

= 60°

Therefore a regular hexagon must be rotated through a minimum angle of 60°.

(i) From the figure it is observed that EF is the intersection between faces EFGH and EFBA.

The edge EF is highlighted in figure:

(ii) From the figure it is observed that faces EFBA and FBCG intersect at edge FB.

Faces are highlighted in the given figure:

(iii) The three faces which form the vertex A are ABFE, ABCD and ADHE.

Faces are highlighted in the given figure:

(iv) Vertex D is formed by the faces ABCD, ADHE and CDHG.

D is highlighted in given figure:

(v) The edges which are parallel to edge AB are:

CD, EF and HG.

The edges are highlighted in given figure:

(vi) The edges that are neither parallel nor perpendicular to edge BC are AE, EF, GH and HD.

The edges are highlighted in given figure:

(vii) The edges that are perpendicular to edge AB are as follows AE, BF, AD and BC.

The edges are highlighted in given figure:

(viii) A, B, G and H are four vertices that do not all lie in one plane.

The vertices are highlighted in given figure:

The required net is as follows:

(i) The required net is as follows:

(ii) The required net is as follows:

(iii) The required net is as follows:

The required net is as follows:

The isometric view of cuboid is as follows:

(i) The edge which meets AN is GH.

(ii) The edge which meets DE is DC.

The required net is as follows:

The required net is as follows:

The required net is as follows:

The number of cubes in given figure is 6.

The number of cubes in given figure is 8.

The number of cubes in given figure is 7.

a) The line of symmetry of the figure is as follows:

b) This figure has no lines of symmetry.

c) The line of symmetry of the figure is as follows:

The total number of faces of the figure is 16.

(a) The line of symmetry of the figure is as follows:

(b) This figure has no lines of symmetry.

(c) The line of symmetry of the figure is as follows:

Figures which have rotational symmetry are as follows:

a, c, d, e and f.

Figure b has no rotational symmetry.

(a) (b)

(c) (d)

(e) (f)

Figures which have rotational symmetry are as follows:

a, b and d.

Figure c has no rotational symmetry.

The above figure does not show rotational symmetry as one part of the figure is not coloured while the other three parts are darken.

Japan’s flag has two line of symmetry which is as shown below:

Only two figures a and c have both line and rotational symmetry.

For figure a,

Also for the rotational symmetry,

Order = 360° / number of sides

= 360° / 8

= 45°

For figure c,

Also for the rotational symmetry,

Order = 360° / number of sides

= 360° / 10

= 36°

(a) The line of symmetry of this figure is as follows:

(b) This figure do not have a line symmetry.

(c) The line of symmetry of this figure is as follows:

(d) This figure do not have a line symmetry.

The capital letters of English alphabet have no line of symmetry are as follows:

F, G, J, L, N, R, Q, S and Z.