Question

Draw a line segment of length 7.6 cm and divide it in the ratio 5:8. Measure both parts.

Hint:

The accuracy of the geometric construction depends on the precision with which the parallel line is drawn. Using a set square or constructing equal corresponding angles carefully will give a better result.

Answer 1

Step 1: Understanding the Concept:
This is a geometric construction problem that uses the concept of similar triangles (based on the Basic Proportionality Theorem) to divide a line segment into a given ratio. We also need to calculate the theoretical lengths of the parts to verify the measurement.

Step 2: Steps of Construction:
\begin{enumerate} \item Draw a line segment AB of length 7.6 cm using a ruler. \item Draw a ray AX from point A, making any acute angle with AB (e.g., \(\angle BAX\)). \item The given ratio is 5:8. The sum of the ratio parts is \(5+8=13\). Using a compass, mark 13 equidistant points on the ray AX, starting from A. Label them \(A_1, A_2, \dots, A_{13}\) such that \(AA_1 = A_1A_2 = \dots = A_{12}A_{13}\). \item Join the last point, \(A_{13}\), to point B to form the line segment \(A_{13}B\). \item From the 5th point on the ray, \(A_5\), draw a line parallel to \(A_{13}B\). This can be done by constructing an angle at \(A_5\) equal to \(\angle AA_{13}B\). This parallel line intersects the original line segment AB at a point C. \item The point C divides the line segment AB in the ratio 5:8. That is, \(AC:CB = 5:8\). \end{enumerate}

Step 3: Measurement and Calculation:
By Measurement:
Using a ruler, measure the lengths of AC and CB. You will find that AC is approximately 2.9 cm and CB is approximately 4.7 cm.
By Calculation:
Total length = 7.6 cm.
Total number of parts in the ratio = 5 + 8 = 13.
Length of the first part (AC) = \(\left(\frac{5}{13}\right) \times 7.6 = \frac{38}{13} \approx 2.923\) cm.
Length of the second part (CB) = \(\left(\frac{8}{13}\right) \times 7.6 = \frac{60.8}{13} \approx 4.677\) cm.

Step 4: Final Answer:
The line segment is divided at point C. On measuring, the two parts AC and CB are found to be approximately 2.9 cm and 4.7 cm, which is consistent with the calculated values of 2.92 cm and 4.68 cm.