Question

assignment: page 95
dilate each triangle with p as the center of dilation and the given scale factor.
ⓐ scale factor of 3
image of point p and triangle with vertices a, c, b
ⓑ scale factor of (\frac{1}{3})
image of point p and right triangle with vertices a, c, b

Explanation:

Part (a) - Scale Factor of 3

Dilation with a center \( P \) and scale factor \( k \) means we find the image of each vertex ( \( A \), \( B \), \( C \)) by drawing a line from \( P \) through the vertex, then extending (or shortening) that line segment so that the distance from \( P \) to the image vertex is \( k \) times the distance from \( P \) to the original vertex.

Step 1: Locate Center and Vertices

Identify center \( P \) and vertices \( A \), \( B \), \( C \) of the triangle.

Step 2: Draw Rays from \( P \)

Draw rays \( \overrightarrow{PA} \), \( \overrightarrow{PB} \), and \( \overrightarrow{PC} \).

Step 3: Scale Distances by 3

For each vertex:

• Measure the distance from \( P \) to \( A \) (let’s call it \( d_{PA} \)). The image \( A' \) lies on \( \overrightarrow{PA} \) such that \( PA' = 3 \cdot PA \).
• Similarly, \( PB' = 3 \cdot PB \) (image \( B' \) on \( \overrightarrow{PB} \)) and \( PC' = 3 \cdot PC \) (image \( C' \) on \( \overrightarrow{PC} \)).

Step 4: Connect Image Vertices

Connect \( A' \), \( B' \), and \( C' \) to form the dilated triangle.

Part (b) - Scale Factor of \( \frac{1}{3} \)

Dilation with scale factor \( \frac{1}{3} \) (a reduction) means the distance from \( P \) to each image vertex is \( \frac{1}{3} \) of the original distance from \( P \) to the vertex.

Step 1: Locate Center and Vertices

Identify center \( P \) and vertices \( A \), \( B \), \( C \) of the triangle.

Step 2: Draw Rays from \( P \)

Draw rays \( \overrightarrow{PA} \), \( \overrightarrow{PB} \), and \( \overrightarrow{PC} \).

Step 3: Scale Distances by \( \frac{1}{3} \)

For each vertex:

• Measure the distance from \( P \) to \( A \) ( \( d_{PA} \) ). The image \( A' \) lies on \( \overrightarrow{PA} \) such that \( PA' = \frac{1}{3} \cdot PA \).
• Similarly, \( PB' = \frac{1}{3} \cdot PB \) (image \( B' \) on \( \overrightarrow{PB} \)) and \( PC' = \frac{1}{3} \cdot PC \) (image \( C' \) on \( \overrightarrow{PC} \)).

Step 4: Connect Image Vertices

Connect \( A' \), \( B' \), and \( C' \) to form the dilated (reduced) triangle.

Final Answer (Visual Description):

• For part (a), the dilated triangle \( A'B'C' \) is larger than \( ABC \), with sides 3 times as long, and \( A' \), \( B' \), \( C' \) collinear with \( P \) and \( A \), \( B \), \( C \) respectively.
• For part (b), the dilated triangle \( A'B'C' \) is smaller than \( ABC \), with sides \( \frac{1}{3} \) as long, and \( A' \), \( B' \), \( C' \) collinear with \( P \) and \( A \), \( B \), \( C \) respectively.

(Note: Since this is a drawing task, the key is to apply the dilation rules to the coordinates/positions of \( P \), \( A \), \( B \), \( C \) as shown in the diagram.)

Answer:

Dilation with scale factor \( \frac{1}{3} \) (a reduction) means the distance from \( P \) to each image vertex is \( \frac{1}{3} \) of the original distance from \( P \) to the vertex.

Step 1: Locate Center and Vertices

Identify center \( P \) and vertices \( A \), \( B \), \( C \) of the triangle.

Step 2: Draw Rays from \( P \)

Draw rays \( \overrightarrow{PA} \), \( \overrightarrow{PB} \), and \( \overrightarrow{PC} \).

Step 3: Scale Distances by \( \frac{1}{3} \)

For each vertex:

• Measure the distance from \( P \) to \( A \) ( \( d_{PA} \) ). The image \( A' \) lies on \( \overrightarrow{PA} \) such that \( PA' = \frac{1}{3} \cdot PA \).
• Similarly, \( PB' = \frac{1}{3} \cdot PB \) (image \( B' \) on \( \overrightarrow{PB} \)) and \( PC' = \frac{1}{3} \cdot PC \) (image \( C' \) on \( \overrightarrow{PC} \)).

Step 4: Connect Image Vertices

Connect \( A' \), \( B' \), and \( C' \) to form the dilated (reduced) triangle.

Final Answer (Visual Description):

• For part (a), the dilated triangle \( A'B'C' \) is larger than \( ABC \), with sides 3 times as long, and \( A' \), \( B' \), \( C' \) collinear with \( P \) and \( A \), \( B \), \( C \) respectively.
• For part (b), the dilated triangle \( A'B'C' \) is smaller than \( ABC \), with sides \( \frac{1}{3} \) as long, and \( A' \), \( B' \), \( C' \) collinear with \( P \) and \( A \), \( B \), \( C \) respectively.

(Note: Since this is a drawing task, the key is to apply the dilation rules to the coordinates/positions of \( P \), \( A \), \( B \), \( C \) as shown in the diagram.)