Question

scale factor of (\frac{1}{4})

Explanation:

To perform a dilation with a scale factor of \(\frac{1}{4}\) centered at point \(P\) for triangle \(ABC\), follow these steps:

Step 1: Identify Key Points

We have the center of dilation \(P\) and the vertices of the triangle: \(A\), \(B\), and \(C\).

Step 2: Dilate Point \(A\)

1. Draw a line from \(P\) through \(A\).
2. Measure the distance from \(P\) to \(A\), then multiply it by \(\frac{1}{4}\) to find the new distance \(PA' = \frac{1}{4} \cdot PA\).
3. Mark the new point \(A'\) on the line \(PA\) such that \(PA' = \frac{1}{4}PA\).

Step 3: Dilate Point \(B\)

1. Draw a line from \(P\) through \(B\).
2. Measure the distance from \(P\) to \(B\), then multiply it by \(\frac{1}{4}\) to find \(PB' = \frac{1}{4} \cdot PB\).
3. Mark \(B'\) on the line \(PB\) such that \(PB' = \frac{1}{4}PB\).

Step 4: Dilate Point \(C\)

1. Draw a line from \(P\) through \(C\).
2. Measure the distance from \(P\) to \(C\), then multiply it by \(\frac{1}{4}\) to find \(PC' = \frac{1}{4} \cdot PC\).
3. Mark \(C'\) on the line \(PC\) such that \(PC' = \frac{1}{4}PC\).

Step 5: Form the Dilated Triangle

Connect the new points \(A'\), \(B'\), and \(C'\) to form the dilated triangle \(A'B'C'\) with a scale factor of \(\frac{1}{4}\) centered at \(P\).

Final Result

The dilated triangle \(A'B'C'\) will be smaller than \(ABC\) (since the scale factor \(\frac{1}{4} < 1\)) and similar in shape, with all sides \(\frac{1}{4}\) the length of the corresponding sides of \(ABC\), and all lines from \(P\) to the vertices of \(A'B'C'\) being \(\frac{1}{4}\) the length of the lines from \(P\) to the vertices of \(ABC\).

Answer:

To perform a dilation with a scale factor of \(\frac{1}{4}\) centered at point \(P\) for triangle \(ABC\), follow these steps:

Step 1: Identify Key Points

We have the center of dilation \(P\) and the vertices of the triangle: \(A\), \(B\), and \(C\).

Step 2: Dilate Point \(A\)

1. Draw a line from \(P\) through \(A\).
2. Measure the distance from \(P\) to \(A\), then multiply it by \(\frac{1}{4}\) to find the new distance \(PA' = \frac{1}{4} \cdot PA\).
3. Mark the new point \(A'\) on the line \(PA\) such that \(PA' = \frac{1}{4}PA\).

Step 3: Dilate Point \(B\)

1. Draw a line from \(P\) through \(B\).
2. Measure the distance from \(P\) to \(B\), then multiply it by \(\frac{1}{4}\) to find \(PB' = \frac{1}{4} \cdot PB\).
3. Mark \(B'\) on the line \(PB\) such that \(PB' = \frac{1}{4}PB\).

Step 4: Dilate Point \(C\)

1. Draw a line from \(P\) through \(C\).
2. Measure the distance from \(P\) to \(C\), then multiply it by \(\frac{1}{4}\) to find \(PC' = \frac{1}{4} \cdot PC\).
3. Mark \(C'\) on the line \(PC\) such that \(PC' = \frac{1}{4}PC\).

Step 5: Form the Dilated Triangle

Connect the new points \(A'\), \(B'\), and \(C'\) to form the dilated triangle \(A'B'C'\) with a scale factor of \(\frac{1}{4}\) centered at \(P\).

Final Result

The dilated triangle \(A'B'C'\) will be smaller than \(ABC\) (since the scale factor \(\frac{1}{4} < 1\)) and similar in shape, with all sides \(\frac{1}{4}\) the length of the corresponding sides of \(ABC\), and all lines from \(P\) to the vertices of \(A'B'C'\) being \(\frac{1}{4}\) the length of the lines from \(P\) to the vertices of \(ABC\).