Questions: True or false: When point B is 1/4 of the distance from point A to point C, then point B partitions AC in segments with lengths in a ratio of I: 4. True False

True or false: When point B is 1/4 of the distance from point A to point C, then point B partitions AC in segments with lengths in a ratio of I: 4.
True
False

Transcript text: True or false: When point $B$ is $\frac{1}{4}$ of the distance from point $A$ to point $C$, then point $B$ partitions $\overline{A C}$ in segments with lengths in a ratio of $\mathrm{I}: 4$. True False

Solution

Solution Steps

To determine if the statement is true or false, we need to understand the concept of partitioning a line segment. If point $B$ is $\frac{1}{4}$ of the distance from point $A$ to point $C$, it means that $B$ divides $\overline{AC}$ into two segments where one segment is $\frac{1}{4}$ of the total length and the other segment is $\frac{3}{4}$ of the total length. We need to check if these segments have lengths in a ratio of $1:4$.

Step 1: Understanding the Problem

We are given that point $ B $ is $ \frac{1}{4} $ of the distance from point $ A $ to point $ C $. This means that the length of segment $ \overline{AB} $ is $ \frac{1}{4} $ of the total length $ \overline{AC} $, while the length of segment $ \overline{BC} $ is $ \frac{3}{4} $ of the total length.

Step 2: Calculating Segment Lengths

Assuming the total length $ \overline{AC} = 1 $:

Length of segment $ \overline{AB} = \frac{1}{4} $
Length of segment $ \overline{BC} = \frac{3}{4} $

Step 3: Finding the Ratio

To find the ratio of the lengths of segments $ \overline{AB} $ to $ \overline{BC} $: \[ \text{Ratio} = \frac{\overline{AB}}{\overline{BC}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3} \]

Step 4: Comparing with the Given Ratio

The problem states that the segments should be in a ratio of $ 1:4 $. However, we found that the actual ratio is $ \frac{1}{3} $, which does not equal $ \frac{1}{4} $.