CBSE Class 7 Maths Chapter 12: Practical Geometry, Symmetry, and Visualizing Solid Shapes - NCERT Solutions

To determine if a triangle can be constructed with given side lengths, we must apply the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's check each option:

1. 1.8 cm + 2.6 cm = 4.4 cm. This sum is not greater than the third side (4.4 cm), so a triangle cannot be formed.
2. 2 cm + 3 cm = 5 cm. This sum (5 cm) is greater than the third side (4 cm). Also, 2 cm + 4 cm = 6 cm > 3 cm, and 3 cm + 4 cm = 7 cm > 2 cm. All conditions are satisfied.
3. 2.4 cm + 2.4 cm = 4.8 cm. This sum is not greater than the third side (6.4 cm), so a triangle cannot be formed.
4. 3.2 cm + 2.3 cm = 5.5 cm. This sum is not greater than the third side (5.5 cm), so a triangle cannot be formed.

Therefore, only option (b) satisfies the conditions for constructing a triangle.

Answer: (b)

The sum of the angles in any triangle is always $180^\{\backslash circ\}$. For a triangle to be constructible with two given angles, the sum of these two angles must be less than $180^\{\backslash circ\}$. This is because the third angle would then be positive ($180^\{\backslash circ\}$ - sum of two angles), which is a requirement for a valid angle in a triangle.

Let's examine each option:

1. $110^\{\backslash circ\}\; +\; 40^\{\backslash circ\}\; =\; 150^\{\backslash circ\}$. Since , a third angle of $180^\{\backslash circ\}\; -\; 150^\{\backslash circ\}\; =\; 30^\{\backslash circ\}$ can be formed, making this a valid triangle construction.
2. $70^\{\backslash circ\}\; +\; 115^\{\backslash circ\}\; =\; 185^\{\backslash circ\}$. This sum is greater than $180^\{\backslash circ\}$, so a triangle cannot be formed.
3. $135^\{\backslash circ\}\; +\; 45^\{\backslash circ\}\; =\; 180^\{\backslash circ\}$. This sum equals $180^\{\backslash circ\}$, which would mean the third angle is $0^\{\backslash circ\}$, not forming a triangle.
4. $90^\{\backslash circ\}\; +\; 90^\{\backslash circ\}\; =\; 180^\{\backslash circ\}$. This sum equals $180^\{\backslash circ\}$, meaning the third angle would be $0^\{\backslash circ\}$, which does not form a triangle.

Therefore, the only pair of angles that can be used to construct a triangle is $110^\{\backslash circ\}$ and $40^\{\backslash circ\}$.

Answer: (a)

A line of symmetry is a line that divides a figure into two identical halves that are mirror images of each other. By visually inspecting the figure (Fig. 12.13), which appears to be a regular hexagon, we can identify lines of symmetry. These lines pass through opposite vertices and through the midpoints of opposite sides. Counting these lines, we find there are 6 such lines.

Answer: (c)

Observing Fig. 12.14, we need to identify lines that divide the figure into two congruent, mirror-image halves. The figure shown has three lines of symmetry. These lines likely pass through vertices and bisect opposite sides, characteristic of shapes like an equilateral triangle or a three-pointed star.

Answer: (b)

Rotational symmetry describes how many times a figure fits onto itself during one complete turn (360 degrees) around its center. The order of rotational symmetry is the count of these times. For the figure shown in Fig. 12.15, it aligns with itself 6 times during a full rotation. This indicates an order of rotational symmetry of 6.

Answer: (c)

The order of rotational symmetry refers to the number of times a figure coincides with its original position during a 360-degree rotation. By examining Fig. 12.16, we can see that the figure fits onto itself exactly 2 times during a full rotation. Therefore, the order of rotational symmetry for this figure is 2.

Answer: (b)