Question

1. which of the following statements are true? select all that apply. a the length of $overline{ab}$ is - 3. b $bc = |6 - 1|$ c $ab+ac = bc$ d $ab + bc=ac$ 2. points a, b, and c are collinear, and b lies between a and c. if $ac = 48$, $ab = 2x + 2$, and $bc = 3x + 6$, what is bc? 3. in the figure shown, $ab = 8$ and $ad = 5$. what is bc? 4. what is the mid - point of $overline{ab}$? 5. what is the length of $overline{ab}$? round your answer to the nearest hundredth

Explanation:

Step1: Analyze statement A for question 1

Length cannot be negative, so A is false.

Step2: Analyze statement B for question 1

If \(B\) is at \(1\) and \(C\) is at \(6\) on the number - line, the distance \(BC=\vert6 - 1\vert\), so B is true.

Step3: Analyze statement C for question 1

By the segment - addition postulate, \(AB+BC = AC\), not \(AB + AC=BC\), so C is false.

Step4: Analyze statement D for question 1

By the segment - addition postulate, for collinear points \(A\), \(B\), \(C\) with \(B\) between \(A\) and \(C\), \(AB + BC=AC\), so D is true.

Step5: Solve for \(x\) in question 2

Since \(AB + BC=AC\) and \(AC = 48\), \(AB=2x + 2\), \(BC=3x + 6\), we have \((2x + 2)+(3x + 6)=48\). Combine like terms: \(5x+8 = 48\). Subtract 8 from both sides: \(5x=40\), then \(x = 8\).

Step6: Find \(BC\) in question 2

Substitute \(x = 8\) into \(BC=3x + 6\), so \(BC=3\times8+6=30\).

Step7: Analyze question 3

In a rectangle, opposite sides are equal. If \(AB = 8\) and \(AD = 5\), and \(BC\) is opposite to \(AD\), then \(BC = 5\).

Answer:

1. B. \(BC=\vert6 - 1\vert\), D. \(AB + BC=AC\)
2. \(30\)
3. \(5\)