Answered

The distance from point $A$ to $C$ is 58.

Complete the proof to show that $AB = 7$.




Statements		Reasons

$A C=58$		Given

$A B=3 x-5$		1

$A C=8 x+19$		Given

$A B+B C=A C$		2

$3 x-5+8 x+19=58$		Substitution Property

$11 x+14=58$		Simplification

$11 x=44$		\_\_\_\_ 3 \_\_\_\_

$x=4$		Division Property of Equality

$A B=3(4)-5$		\_\_\_\_ 4 \_\_\_\_

$A B=7$		Simplification






Blank 1: [Choose]

Blank 2: [Choose]

Blank 3: [Choose]

Blank 4: [Choose]

Verified Answer :

  • Blank 1 is 'Given' because the expression for AB is stated as part of the problem.
  • Blank 2 is 'Segment Addition Postulate' because it relates the lengths of segment parts to the whole.
  • Blank 3 is 'Subtraction Property of Equality' because 14 is subtracted from both sides to isolate the variable term.
  • Blank 4 is 'Substitution Property of Equality' because the solved value of \(x\) is used in the expression for AB.

Final answers:

  • Blank 1: \(\boxed{\text{Given}}\)
  • Blank 2: \(\boxed{\text{Segment Addition Postulate}}\)
  • Blank 3: \(\boxed{\text{Subtraction Property of Equality}}\)
  • Blank 4: \(\boxed{\text{Substitution Property of Equality}}\)

Explanation

  1. Analyze the two-column proof and identify reasons.

The problem requires identifying the justifications for specific steps in a two-column proof.
The given data is:

  • Statement: \(AC = 58\), Reason: Given.
  • Statement: \(AB = 3x - 5\), Reason: Blank 1. This is a definition provided to model the length \(AB\), so it is Given.
  • Statement: \(AC = 8x + 19\), Reason: Given.
  • Statement: \(AB + BC = AC\), Reason: Blank 2. This represents the Segment Addition Postulate, which states that for any three collinear points \(A, B,\) and \(C\), if \(B\) is between \(A\) and \(C\), then \(AB + BC = AC\).
  • Statement: \(3x - 5 + 8x + 19 = 58\), Reason: Substitution Property.
  • Statement: \(11x + 14 = 58\), Reason: Simplification.
  • Statement: \(11x = 44\), Reason: Blank 3. This follows from subtracting 14 from both sides of the equation \(11x + 14 = 58\), which is the Subtraction Property of Equality.
  • Statement: \(x = 4\), Reason: Division Property of Equality.
  • Statement: \(AB = 3(4) - 5\), Reason: Blank 4. This step substitutes the value \(x = 4\) into the previously defined expression for \(AB = 3x - 5\), which is the Substitution Property of Equality.
  • Statement: \(AB = 7\), Reason: Simplification.
  1. Final results for the blanks.

Based on the analysis, the blanks correspond to the following reasons:

  • Blank 1: 'Given'
  • Blank 2: 'Segment Addition Postulate'
  • Blank 3: 'Subtraction Property of Equality'
  • Blank 4: 'Substitution Property of Equality'

We have verified that substituting \(x=4\) into \(AB = 3(4)-5\) results in \(AB = 12-5 = 7\).
The final answers for the blanks are:
Blank 1: Given
Blank 2: Segment Addition Postulate
Blank 3: Subtraction Property of Equality
Blank 4: Substitution Property of Equality.

Examples

A common use of two-column proofs in real life is in engineering and construction, where every measurement and adjustment must be logically justified. For example, when confirming that a support beam is the correct length, an engineer might use the Segment Addition Postulate to ensure that the sum of the individual segments of the beam structure correctly accounts for the total length required for the building's design. \(\boxed{\text{Given}} \rightarrow \boxed{\text{Segment Addition Postulate}} \rightarrow \dots \rightarrow \text{Final structural stability!}\)