Question
The points \( \mathrm{C}, \mathrm{D}, \mathrm{E} \) and F all lie on the same line segment, in that order, such that the ratio of \( C D: D E: E F \) is equal to \( 1: 2: 4 \). If \( C F=7 \), find \( C D \). Answer Attempt 1 out of 3 \( C D=\square \)
Ask by Weston Delgado. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( CD = 1 \)
Solution
Let’s denote:
- \( CD = x \)
- \( DE = 2x \) (since the ratio is 1:2:4)
- \( EF = 4x \)
The total length \( CF \) is the sum of these segments:
\[
CF = CD + DE + EF = x + 2x + 4x = 7x
\]
Given that \( CF = 7 \), we can set up the equation:
\[
7x = 7
\]
Solving for \( x \):
\[
x = 1
\]
Therefore, the length of \( CD \) is:
\[
CD = 1
\]
**Answer:**
\( CD = 1 \)
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Simplify this solution Extra Insights
To solve this problem, let's denote the lengths of segments \( CD \), \( DE \), and \( EF \) as \( x \), \( 2x \), and \( 4x \) respectively, according to the given ratio \( 1:2:4 \). Now, we can find the total length \( CF \) as the sum of the lengths of all segments: \[ CF = CD + DE + EF = x + 2x + 4x = 7x. \] We are told that \( CF = 7 \), so we can set up the equation: \[ 7x = 7. \] To find \( x \), we can divide both sides by 7: \[ x = 1. \] Now, substituting back to find \( CD \): \[ CD = x = 1. \] Thus, the length of \( CD \) is \( \boxed{1} \).
