Question

triangle △abc is the result of dilating △abc about point p by a scale factor of 3. determine whether each claim about the properties of △abc and △abc is true or false. ab and ab are on the same line. true/false ac and ac are on distinct parallel lines. true/false

Explanation:

Step1: Recall Dilation Properties

Dilation about a point \( P \) means that lines through \( P \) and corresponding points (e.g., \( A \) and \( A' \), \( B \) and \( B' \), \( C \) and \( C' \)) are collinear, and the image of a line segment (like \( AB \)) under dilation about \( P \) will be a segment \( A'B' \) such that \( AB \parallel A'B' \) (or colinear if \( P \) is on the line of \( AB \)). Also, the lines containing \( AC \) and \( A'C' \) pass through \( P \), so they are the same line (not distinct parallel lines).

Step2: Analyze \( AB \) and \( A'B' \)

Since dilation about \( P \), the vector from \( P \) to \( A \) and \( P \) to \( A' \) are scalar multiples (scale factor 3), same for \( P \) to \( B \) and \( P \) to \( B' \). So the line through \( A \), \( B \) and the line through \( A' \), \( B' \) are either parallel or colinear. But since \( P \) is not on \( AB \) (from the diagram), \( AB \) and \( A'B' \) are parallel? Wait, no—wait, if you dilate a segment about a point not on the segment, the image is parallel. Wait, but the first claim: "AB and A'B' are on the same line"—wait, maybe I misread. Wait, the diagram: let's check. Wait, maybe \( P \) is on the line of \( AB \)? Wait, no, the diagram shows \( P \) on \( AC \). Wait, no, the first claim: "AB and A'B' are on the same line"—wait, dilation: when you dilate a segment \( AB \) about \( P \), the image \( A'B' \) is such that \( \overrightarrow{PA'} = 3\overrightarrow{PA} \) and \( \overrightarrow{PB'} = 3\overrightarrow{PB} \). So the slope of \( AB \) and \( A'B' \) is the same, so they are parallel. But if \( P \) is not on \( AB \), they are parallel, not colinear. Wait, maybe the diagram shows \( P \) not on \( AB \), so \( AB \) and \( A'B' \) are parallel, not same line? Wait, no—wait, maybe I made a mistake. Wait, the first claim: "AB and A'B' are on the same line"—let's re-examine. Wait, maybe the problem has a typo, but according to dilation: if \( P \) is on the line of \( AB \), then \( AB \) and \( A'B' \) are colinear. But from the diagram, \( P \) is on \( AC \), not \( AB \). So \( AB \) and \( A'B' \) should be parallel, not same line? Wait, no—wait, no, dilation preserves parallelism. So the line \( AB \) and line \( A'B' \) are parallel. But the claim is "on the same line"—so that would be false? Wait, no, maybe I misinterpret. Wait, let's check the second claim: "AC and A'C' are on distinct parallel lines"—but \( AC \) and \( A'C' \) pass through \( P \), so they are the same line, so the claim is false.

Wait, maybe I messed up the first claim. Let's re-express: Dilation about \( P \): the line through \( A \) and \( A' \) passes through \( P \), line through \( B \) and \( B' \) passes through \( P \). So the lines \( PA \) and \( PA' \) are colinear, \( PB \) and \( PB' \) are colinear. So the slope of \( AB \): \( m_{AB} = \frac{y_B - y_A}{x_B - x_A} \), slope of \( A'B' \): \( m_{A'B'} = \frac{y_{B'} - y_{A'}}{x_{B'} - x_{A'}} \). Since \( y_{A'} = y_A + 3(y_A - y_P) \)? No, dilation about \( P \): \( A' = P + 3(A - P) = 3A - 2P \). Similarly for \( B' \). So the vector \( A'B' = B' - A' = 3(B - A) \), so \( A'B' \) is a scalar multiple of \( AB \), so they are parallel. If \( AB \) and \( A'B' \) are parallel, are they on the same line? Only if \( P \) is on \( AB \), which it's not (from diagram, \( P \) is on \( AC \)). So \( AB \) and \( A'B' \) are parallel, not same line—so the first claim "AB and A'B' are on the same line" is false? Wait, but maybe the diagram shows \( P \) on \( AB \)? No, the diagram shows \( P \) on \( AC \). Wait, maybe the problem's first claim is "AB and A'B' are parallel" but it says "on the same line"—so that's false. But wait, maybe I'm wrong. Let's check the second claim: "AC and A'C' are on distinct parallel lines"—but \( AC \) and \( A'C' \) share the point \( P \), so they are the same line (since two points determine a line, and \( P \), \( A \), \( C \) are on one line, \( P \), \( A' \), \( C' \) are on the same line). So they are not distinct parallel lines—so that claim is false.

Wait, maybe the first claim is true? Wait, no—if \( AB \) and \( A'B' \) are parallel, but not same line. Wait, maybe the problem has a different diagram. Alternatively, maybe the first claim is "AB and A'B' are parallel" (same line would be colinear, but parallel includes colinear? No, parallel lines are either distinct or coincident. Wait, in some definitions, parallel lines can be coincident (same line). But usually, distinct parallel lines are non-coincident. Wait, the first claim: "AB and A'B' are on the same line"—if dilation about \( P \), and \( P \) is not on \( AB \), then \( AB \) and \( A'B' \) are parallel (coincident only if \( P \) is on \( AB \)). So if \( P \) is not on \( AB \), they are distinct parallel lines? No, wait, no—dilation of a segment not through the center: the image is parallel. So \( AB \parallel A'B' \), and if \( P \) is not on \( AB \), then \( AB \) and \( A'B' \) are distinct parallel lines, not same line. Wait, I'm confused. Let's take coordinates. Suppose \( P \) is at (let's say) (2,2), \( A \) at (3,3), \( B \) at (5,3). Then \( A' = 3*(3,3) - 2*(2,2) = (9 - 4, 9 - 4) = (5,5) \), \( B' = 3*(5,3) - 2*(2,2) = (15 - 4, 9 - 4) = (11,5) \). Then \( AB \) is from (3,3) to (5,3), horizontal line \( y=3 \). \( A'B' \) is from (5,5) to (11,5), horizontal line \( y=5 \). So they are parallel, distinct lines—so not same line. So first claim "AB and A'B' are on the same line" is false? But wait, maybe the diagram is different. Alternatively, maybe the first claim is "AB and A'B' are parallel" (but the problem says "on the same line"). Wait, the problem's first claim: "AB and A'B' are on the same line"—so that's false. Second claim: "AC and A'C' are on distinct parallel lines"—but \( AC \) and \( A'C' \) pass through \( P \), so they are the same line, so the claim is false.

Wait, maybe I made a mistake. Let's re-express dilation properties:

- Dilation about a point \( P \) preserves the direction of lines: the line through \( A \) and \( A' \) is the same as the line through \( P \) and \( A \) (since \( A' \) is on \( \overleftrightarrow{PA} \)).

- The image of a line segment \( AB \) under dilation about \( P \) is a segment \( A'B' \) such that \( AB \parallel A'B' \) (if \( P \) is not on \( AB \)) or \( AB \) and \( A'B' \) are colinear (if \( P \) is on \( AB \)).

So for the first claim: "AB and A'B' are on the same line"—this is true only if \( P \) is on \( AB \). From the diagram, \( P \) is on \( AC \), not \( AB \), so \( AB \parallel A'B' \) (distinct lines), so the claim is false? Wait, no—wait, in the coordinate example above, \( AB \) and \( A'B' \) are parallel (same slope) but different lines (different \( y \)-intercepts). So they are not on the same line. So first claim: False.

Second claim: "AC and A'C' are on distinct parallel lines"—but \( AC \) and \( A'C' \) are on the same line (since \( P \), \( A \), \( C \), \( A' \), \( C' \) are colinear), so they are not distinct parallel lines (they are the same line), so the claim is False.

Answer:

• "AB and \( A'B' \) are on the same line": False
• "AC and \( A'C' \) are on distinct parallel lines": False