QUESTION IMAGE
Question
- fill in the blanks to complete the proof. (segment addition) given: ( gd = 3x - 4 ), ( dp = 13 ), ( gp = 27 ). prove: ( x = 6 ). table with columns “statements” (rows: ( gd = 3x - 4 ), ( dp = 13 ), ( gp = 27 ), ( gd + dp = gp ), ( 3x + 9 = 27 ), blank, ( x = 6 )) and “reasons” (rows: given, given, given, blank, blank, subtraction property, division property). diagram: segment ( gp ) with point ( d ) between ( g ) (left) and ( p ) (right).
Step1: Identify the segment addition postulate
The segment addition postulate states that if \( D \) is between \( G \) and \( P \), then \( GD + DP = GP \). So the reason for \( GD + DP = GP \) is the Segment Addition Postulate.
Step2: Substitute the known values
We know \( GD = 3x - 4 \), \( DP = 13 \), and \( GP = 27 \). Substituting these into \( GD + DP = GP \), we get \( (3x - 4)+13 = 27 \). Simplify the left - hand side: \( 3x+(- 4 + 13)=27\), so \( 3x + 9 = 27 \). The reason for this step is Substitution Property (substituting the given values of \( GD \), \( DP \), and \( GP \) into the segment addition equation).
Step3: Use the subtraction property of equality
To isolate the term with \( x \), we subtract 9 from both sides of the equation \( 3x + 9 = 27 \). By the subtraction property of equality (if \( a=b \), then \( a - c=b - c \)), we have \( 3x+9 - 9=27 - 9 \), which simplifies to \( 3x=18 \).
Step4: Use the division property of equality
To solve for \( x \), we divide both sides of the equation \( 3x = 18 \) by 3. By the division property of equality (if \( a=b \), then \( \frac{a}{c}=\frac{b}{c} \) for \( c
eq0 \)), we get \( \frac{3x}{3}=\frac{18}{3} \), so \( x = 6 \).
Filling in the table:
| Statements | Reasons |
|---|---|
| \( GD + DP = GP \) | Segment Addition Postulate |
| \( (3x - 4)+13 = 27 \) | Substitution Property |
| \( 3x + 9 = 27 \) | Simplification (from previous step) |
| \( 3x+9 - 9=27 - 9 \) (i.e., \( 3x = 18 \)) | Subtraction Property of Equality |
| \( \frac{3x}{3}=\frac{18}{3} \) (i.e., \( x = 6 \)) | Division Property of Equality |
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The completed proof table has the following reasons (and corresponding statements as above): For \( GD + DP = GP \), the reason is "Segment Addition Postulate"; for \( (3x - 4)+13 = 27 \), the reason is "Substitution Property"; for \( 3x = 18 \), the reason is "Subtraction Property of Equality"; and the final step to get \( x = 6 \) uses the "Division Property of Equality". The key steps fill in the blanks with the appropriate properties and the derived equations.