Q1:

In each of the following, give the justification of the construction also: Draw a line segment of length 7.6 cm and divide it in the ratio 5 : 8. Measure the two parts.

Given: A line segment $AB$ of length $7.6\text{ cm}$.

To Find: Divide the line segment $AB$ in the ratio $5 : 8$ and measure the lengths of the two parts.

Visual Representation:

Steps of Construction:

Step 1: Draw a line segment $AB = 7.6\text{ cm}$ using a ruler.

Step 2: Draw any ray $AX$ making an acute angle with $AB$.

Step 3: Locate $5 + 8 = 13$ points $A_1, A_2, \dots, A_{13}$ on $AX$ such that $AA_1 = A_1A_2 = \dots = A_{12}A_{13}$.

Step 4: Join $BA_{13}$.

Step 5: Through the point $A_5$, draw a line parallel to $BA_{13}$ (by making an angle equal to $\angle AA_{13}B$ at $A_5$) to intersect $AB$ at a point $P$.

Step 6: The point $P$ divides $AB$ in the ratio $5 : 8$.

Justification:

In $\triangle ABA_{13}$, $A_5P$ is parallel to $A_{13}B$ (by construction).
By the Basic Proportionality Theorem (Thales Theorem), if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Therefore, $\frac{AP}{PB} = \frac{AA_5}{A_5A_{13}}$

Since $AA_5$ contains $5$ equal parts and $A_5A_{13}$ contains $8$ equal parts, we have:

$\frac{AP}{PB} = \frac{5}{8}$

This justifies that point $P$ divides $AB$ in the ratio $5 : 8$.

Measurement:

The total length is $7.6\text{ cm}$. The ratio is $5 : 8$.
Sum of ratio parts = $5 + 8 = 13$.

Length of first part $AP = \frac{5}{13} \times 7.6\text{ cm} \approx 2.92\text{ cm}$.

Length of second part $PB = \frac{8}{13} \times 7.6\text{ cm} \approx 4.68\text{ cm}$.

Final Answer: The line segment is divided into two parts measuring approximately 2.92 cm and 4.68 cm.