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Let's walk through the questions on your geometry worksheet and solve them step by step.

### 1. Points \( P \), \( Q \), and \( R \) are collinear.
**a. Find \( PQ \):**  
From the number line, \( P = -2 \) and \( Q = 0 \). The distance between \( P \) and \( Q \) is:  
\[ PQ = |0 - (-2)| = 2 \]

**b. If \( R \)'s coordinate is 7, why is \( PQ \) not congruent to \( QR \)?**  
We found that \( PQ = 2 \). For \( QR \), the distance is:  
\[ QR = |7 - 0| = 7 \]  
Since \( PQ = 2 \) and \( QR = 7 \), they are not congruent because their lengths are different.

**c. What must the coordinate of \( R \) be in order for \( Q \) to be the midpoint of \( PR \)?**  
If \( Q \) is the midpoint of \( PR \), the distance from \( P \) to \( Q \) must be equal to the distance from \( Q \) to \( R \).  
Thus,  
\[ PQ = QR \]  
Since \( PQ = 2 \), we need \( QR = 2 \) as well. Therefore,  
\[ R = Q + QR = 0 + 2 = 2 \]  
So, the coordinate of \( R \) must be 2.

### 2. Points \( W \), \( X \), and \( Y \) are collinear. \( WY = 25 \) and the ratio of \( WX \) to \( XY \) is 2:3. Find \( WX \).  
Let \( WX = 2k \) and \( XY = 3k \), where \( k \) is a constant.  
Thus,  
\[ WX + XY = WY \]  
Substitute the values:  
\[ 2k + 3k = 25 \]  
\[ 5k = 25 \]  
\[ k = 5 \]  
So,  
\[ WX = 2k = 2(5) = 10 \]

### 3. The lengths of two segments are in the ratio 5:3, and the longer segment exceeds the shorter segment by 14. Find the length of the longer segment.
Let the length of the shorter segment be \( 3x \), and the length of the longer segment be \( 5x \).  
The difference between the longer and shorter segment is 14:  
\[ 5x - 3x = 14 \]  
\[ 2x = 14 \]  
\[ x = 7 \]  
So, the length of the longer segment is:  
\[ 5x = 5(7) = 35 \]

### 4. If \( CD = 5x - 7 \), find the value of \( x \), \( CE \), and \( CD \).  
From the image, \( CE = 2x - 3 \) and \( ED = 3x - 4 \), and it is implied that \( CD = CE + ED \).  
Thus,  
\[ CD = CE + ED \]  
Substitute the expressions:  
\[ 5x - 7 = (2x - 3) + (3x - 4) \]  
Simplify:  
\[ 5x - 7 = 5x - 7 \]  
This is true for all \( x \), so no specific value of \( x \) is given. The values of \( CE \) and \( CD \) are left in terms of \( x \).

### 5. If \( C \) is the midpoint of \( KN \), what is \( KC \)?  
If \( C \) is the midpoint, then \( KC = CN \).  
From the diagram, \( KC = 2x + 10 \) and \( CN = 4x + 1 \).  
Since \( KC = CN \):  
\[ 2x + 10 = 4x + 1 \]  
Solve for \( x \):  
\[ 10 - 1 = 4x - 2x \]  
\[ 9 = 2x \]  
\[ x = \frac{9}{2} = 4.5 \]  
Substitute \( x \) back to find \( KC \):  
\[ KC = 2(4.5) + 10 = 9 + 10 = 19 \]

Let me know if you'd like further explanations on any of these problems or need help with the remaining ones!

Geometry worksheet involving segments, their measures, and collinearity. The questions cover topics such as segment addition postulate, ratios of segments, midpoint calculation, and congruency of segments.

This guide provides step-by-step solutions to common geometry problems involving segment addition postulate, collinearity, midpoints, and ratios of segments. Learn how to solve for missing values and understand key geometry concepts suitable for Grades 7-9.

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