Question

comparing ratios and fractions
explain why partitioning a directed line segment into a ratio of 1:2 is not the same as finding half the length of the directed line segment.

Explanation:

When partitioning a directed line segment in a ratio of \(1:2\), we divide the segment into \(1 + 2=3\) equal parts. The first part (corresponding to the "1" in the ratio) is \(\frac{1}{3}\) of the total length, and the second part (corresponding to the "2" in the ratio) is \(\frac{2}{3}\) of the total length. When finding half the length of the directed line segment, we are dividing it into two equal parts, each of which is \(\frac{1}{2}\) of the total length. Since \(\frac{1}{3}
eq\frac{1}{2}\) and \(\frac{2}{3}
eq\frac{1}{2}\), partitioning in the ratio \(1:2\) is not the same as finding half the length.

Answer:

Partitioning a directed line segment in a ratio of \(1:2\) divides it into 3 equal parts (with the two parts being \(\frac{1}{3}\) and \(\frac{2}{3}\) of the total length), while finding half the length divides it into 2 equal parts (each \(\frac{1}{2}\) of the total length). Since \(\frac{1}{3}
eq\frac{1}{2}\) and \(\frac{2}{3}
eq\frac{1}{2}\), these are not the same.