### Geometry Problem: Collinear Points and Segment Ratios**Problem Statement:**The points C, D, E, and F all lie on the same line segment, in that order, such that the ratio of $CD : DE : EF$ is equal to $4 : 5 : 2$. If $CF = 44$, find $EF$.**Solution:**To solve this problem, we start by letting the lengths of $CD$, $DE$, and $EF$ be represented as $4x$, $5x$, and $2x$ respectively. According to the given information, the total length of $CF$ is equal to the sum of $CD$, $DE$, and $EF$. Therefore, we have the following equation:\[ CF = CD + DE + EF \]Since $CF = 44$, we substitute in the expressions for $CD$, $DE$, and $EF$:\[ 44 = 4x + 5x + 2x \]Combining the terms on the right-hand side, we get:\[ 44 = 11x \]Solving for $x$:\[ x = \frac{44}{11} = 4 \]Now, we need to find the length of $EF$:\[ EF = 2x \]Substitute $x = 4$:\[ EF = 2 \times 4 = 8 \]Thus, the length of $EF$ is $8$.

Answered: The points C, D, E and F all lie on the…

The points C, D, E and F all lie on the same line segment, in that order, such that the ratio of CD : DE : EF is equal to 4 : 5 : 2. If CF = 44, find EF. %3D

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter5: Trigonometric Functions: Right Triangle Approach

Section5.1: Angle Measure

Problem 4E: Object A is travelling along a circle of radius 2, and Object B is travelling along a circle of...

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Question

### Geometry Problem: Collinear Points and Segment Ratios

**Problem Statement:**
The points C, D, E, and F all lie on the same line segment, in that order, such that the ratio of $CD : DE : EF$ is equal to $4 : 5 : 2$. If $CF = 44$, find $EF$.

**Solution:**
To solve this problem, we start by letting the lengths of $CD$, $DE$, and $EF$ be represented as $4x$, $5x$, and $2x$ respectively. According to the given information, the total length of $CF$ is equal to the sum of $CD$, $DE$, and $EF$. Therefore, we have the following equation:

\[
CF = CD + DE + EF
\]

Since $CF = 44$, we substitute in the expressions for $CD$, $DE$, and $EF$:

\[
44 = 4x + 5x + 2x
\]

Combining the terms on the right-hand side, we get:

\[
44 = 11x
\]

Solving for $x$:

\[
x = \frac{44}{11} = 4
\]

Now, we need to find the length of $EF$:

\[
EF = 2x
\]

Substitute $x = 4$:

\[
EF = 2 \times 4 = 8
\]

Thus, the length of $EF$ is $8$.

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