CBSE Class 6 Mathematics Chapter 5: Understanding Elementary Shapes NCERT Solutions

Solution: The primary disadvantage of comparing line segments by mere observation is the high chance of making errors. Our eyes might not be able to accurately perceive very small differences in length, especially if the segments are not placed perfectly side-by-side or if the viewing angle is not ideal. This can lead to incorrect conclusions about which segment is longer or shorter.

Solution: It is generally better to use a divider than a ruler for measuring the length of a line segment because a divider allows for a more precise transfer of length. When using a ruler, the thickness of the ruler itself or the width of the markings can sometimes introduce slight inaccuracies in reading the exact length. A divider, with its sharp points, can be placed precisely at the endpoints of the line segment. The distance between the divider's points can then be measured against a scale or compared with another segment without the potential errors associated with the ruler's physical dimensions.

Solution: Yes, if point C lies strictly between points A and B on a straight line, then the length of the segment AB will be exactly equal to the sum of the lengths of the segments AC and CB. This is a fundamental property known as the Segment Addition Postulate.

Let's consider an example:

Draw a line segment AB. Choose a point C on the segment AB such that it is between A and B.

Suppose we measure the lengths and find:

$$AB\; =\; 6.5\; \backslash text\{\; cm\}$$

$$AC\; =\; 3\; \backslash text\{\; cm\}$$

$$CB\; =\; 3.5\; \backslash text\{\; cm\}$$

Now, let's check the condition:

$$AC\; +\; CB\; =\; 3\; \backslash text\{\; cm\}\; +\; 3.5\; \backslash text\{\; cm\}\; =\; 6.5\; \backslash text\{\; cm\}$$

Comparing this sum with the length of AB, we see that:

$$AC\; +\; CB\; =\; 6.5\; \backslash text\{\; cm\}\; =\; AB$$

Therefore, the statement $AB = AC + CB$ holds true when C lies between A and B.

Solution: We are given three points A, B, and C on a straight line with the following lengths:

$$AB\; =\; 5\; \backslash text\{\; cm\}$$

$$BC\; =\; 3\; \backslash text\{\; cm\}$$

$$AC\; =\; 8\; \backslash text\{\; cm\}$$

According to the Segment Addition Postulate, if the sum of the lengths of two smaller segments equals the length of the largest segment, then the common endpoint of the two smaller segments lies between the other two points. In this case, we observe that:

$$AB\; +\; BC\; =\; 5\; \backslash text\{\; cm\}\; +\; 3\; \backslash text\{\; cm\}\; =\; 8\; \backslash text\{\; cm\}$$

Since $AB + BC = AC$, the point B, which is the common endpoint of segments AB and BC, must lie between points A and C.

Thus, B lies between A and C.

Solution: To verify if D is the midpoint of the line segment AG, we need to check if the distance from A to D is equal to the distance from D to G.

From the given information, the points are arranged consecutively with 1 unit between each adjacent point. Let's count the units:

The segment AD consists of the segments AB, BC, and CD. Each of these segments has a length of 1 unit.

$$AD\; =\; AB\; +\; BC\; +\; CD\; =\; 1\; +\; 1\; +\; 1\; =\; 3\; \backslash text\{\; units\}$$

The segment DG consists of the segments DE, EF, and FG. Each of these segments also has a length of 1 unit.

$$DG\; =\; DE\; +\; EF\; +\; FG\; =\; 1\; +\; 1\; +\; 1\; =\; 3\; \backslash text\{\; units\}$$

Now, we compare the lengths of AD and DG:

$$AD\; =\; 3\; \backslash text\{\; units\}$$

$$DG\; =\; 3\; \backslash text\{\; units\}$$

Since $AD = DG$, point D divides the line segment AG into two equal parts. Therefore, D is the midpoint of AG.

Solution: We are given that B is the midpoint of the line segment AC. By the definition of a midpoint, it divides a segment into two equal parts. Therefore, the length of segment AB is equal to the length of segment BC.

AB = BC \quad \cdots (i)

We are also given that C is the midpoint of the line segment BD. Similarly, this means that the length of segment BC is equal to the length of segment CD.

BC = CD \quad \cdots (ii)

Now, we can use these two equations to establish a relationship between AB and CD. From equation (i), we know $AB = BC$, and from equation (ii), we know $BC = CD$. Since both AB and CD are equal to BC, they must be equal to each other.

By substituting BC from equation (i) into equation (ii) (or vice versa), or by using the transitive property of equality, we can conclude:

AB = CD

Thus, we have shown that $AB = CD$.

Solution: The statement in the question asks to check if the sum of the lengths of any two sides is always *less than* the third side. However, a fundamental property of triangles, known as the Triangle Inequality Theorem, states that the sum of the lengths of any two sides of a triangle must always be *greater than* the length of the third side.

Let's demonstrate with examples of triangles:

Triangle 1:

Sides: 5 cm, 6 cm, 7 cm

Check sums:

• $5 + 6 = 11$ cm, which is greater than 7 cm.
• $5 + 7 = 12$ cm, which is greater than 6 cm.
• $6 + 7 = 13$ cm, which is greater than 5 cm.

Triangle 2:

Sides: 10 cm, 12 cm, 15 cm

Check sums:

• $10 + 12 = 22$ cm, which is greater than 15 cm.
• $10 + 15 = 25$ cm, which is greater than 12 cm.
• $12 + 15 = 27$ cm, which is greater than 10 cm.

Triangle 3:

Sides: 3 cm, 4 cm, 5 cm (a right-angled triangle)

Check sums:

• $3 + 4 = 7$ cm, which is greater than 5 cm.
• $3 + 5 = 8$ cm, which is greater than 4 cm.
• $4 + 5 = 9$ cm, which is greater than 3 cm.

Triangle 4:

Sides: 8 cm, 10 cm, 17 cm

Check sums:

• $8 + 10 = 18$ cm, which is greater than 17 cm.
• $8 + 17 = 25$ cm, which is greater than 10 cm.
• $10 + 17 = 27$ cm, which is greater than 8 cm.

Triangle 5:

Sides: 5.2 cm, 8 cm, 10 cm

Check sums:

• $5.2 + 8 = 13.2$ cm, which is greater than 10 cm.
• $5.2 + 10 = 15.2$ cm, which is greater than 8 cm.
• $8 + 10 = 18$ cm, which is greater than 5.2 cm.

In all these cases, the sum of the lengths of any two sides is always greater than the third side, confirming the Triangle Inequality Theorem. It is impossible to form a triangle where the sum of two sides is less than the third side.