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

Solution: Comparing line segments by just looking at them (mere observation) can be unreliable. The main disadvantage is the chance of error due to improper viewing. Our eyes might not perceive the exact lengths accurately, especially if the segments are very close in length or viewed from different angles.

Solution: It is generally better to use a divider than a ruler for precise measurement of a line segment because:

1. Ruler Accuracy: The thickness of the ruler's markings and the angle at which you view them can cause slight inaccuracies when reading the length.
2. Divider Accuracy: A divider allows you to set its points exactly at the endpoints of the line segment. You can then transfer this precise setting to a ruler or scale without the issue of viewing angle or marking thickness, leading to a more accurate measurement.

Therefore, a divider provides a more accurate measurement of a line segment's length.

Solution: Let's consider a line segment AB and a point C located between A and B. We will measure the lengths:

Let's assume we draw a line segment AB and place a point C in between A and B. After measuring:

Measured length of $AB = 6.5 \text{ cm}$

Measured length of $AC = 3 \text{ cm}$

Measured length of $CB = 3.5 \text{ cm}$

Now, let's check if $AC + CB = AB$:

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

Since $AC + CB = 6.5 \text{ cm}$, which is equal to the measured length of $AB$, the equation $AB = AC + CB$ holds true.

Yes, in this case, $AB = AC + CB$. This demonstrates the segment addition postulate, where if C lies between A and B on a line, the length of the whole segment AB is the sum of the lengths of the segments AC and CB.

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

• $AB = 5 \text{ cm}$
• $BC = 3 \text{ cm}$
• $AC = 8 \text{ cm}$

To determine which point lies between the other two, we check if the sum of the lengths of two segments equals the length of the third segment. This is based on the segment addition postulate.

Let's check the sums:

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

We observe that $AB + BC$ is equal to $AC$ ($8 \text{ cm}$). According to the segment addition postulate, if the sum of the lengths of two smaller segments equals the length of the longest segment, the common endpoint of the two smaller segments is the point lying between the other two.

In this case, the common endpoint of AB and BC is B. Therefore, point B lies between points A and C.

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

From the given arrangement, we can count the units:

• The segment AG consists of segments AB, BC, CD, DE, EF, FG.
• The total length of AG is the sum of the lengths of these smaller segments.
• The distance from A to D ($AD$) covers the segments AB, BC, and CD.
• The distance from D to G ($DG$) covers the segments DE, EF, and FG.

Counting the units:

Length of $AD = AB + BC + CD = 1 \text{ unit} + 1 \text{ unit} + 1 \text{ unit} = 3 \text{ units}$.

Length of $DG = DE + EF + FG = 1 \text{ unit} + 1 \text{ unit} + 1 \text{ unit} = 3 \text{ units}$.

Since $AD = 3 \text{ units}$ and $DG = 3 \text{ units}$, we have $AD = DG$.

Therefore, D is the midpoint of AG because it divides the line segment AG into two equal parts.

Solution: We are given that A, B, C, and D are points on a straight line.

We are also given two conditions:

1. B is the mid-point of AC: This means that B divides the line segment AC into two equal parts. Mathematically, this can be written as:

   $AB\; =\; BC$ ... (i)

2. C is the mid-point of BD: This means that C divides the line segment BD into two equal parts. Mathematically, this can be written as:

   $BC\; =\; CD$ ... (ii)

Now, we need to show that $AB = CD$.

From equation (i), we know that $AB$ is equal to $BC$.

From equation (ii), we know that $BC$ is equal to $CD$.

Since $AB$ is equal to $BC$, and $BC$ is equal to $CD$, we can conclude through the transitive property of equality that $AB$ must be equal to $CD$.

Therefore, $AB = CD$.

Solution: The question asks to check if the sum of the lengths of any two sides of a triangle is always *less than* the third side. This statement is actually contrary to a fundamental property of triangles.

The actual property is the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

Let's demonstrate with examples:

Example 1: Consider a triangle with sides 5 cm, 8 cm, and 10 cm.

• $5 + 8 = 13 \text{ cm}$, which is greater than $10 \text{ cm}$.
• $5 + 10 = 15 \text{ cm}$, which is greater than $8 \text{ cm}$.
• $8 + 10 = 18 \text{ cm}$, which is greater than $5 \text{ cm}$.

This triangle satisfies the Triangle Inequality Theorem.

Example 2: Consider a triangle with sides 17 cm, 15 cm, and 9 cm.

• $17 + 15 = 32 \text{ cm}$, which is greater than $9 \text{ cm}$.
• $17 + 9 = 26 \text{ cm}$, which is greater than $15 \text{ cm}$.
• $15 + 9 = 24 \text{ cm}$, which is greater than $17 \text{ cm}$.

This triangle also satisfies the Triangle Inequality Theorem.

Conclusion: When you draw and measure five different triangles, you will find that in every case, the sum of the lengths of any two sides is always greater than the third side, not less than. The statement in the question is incorrect based on geometric principles.