Question

if triangle abc is dilated about point a with a scale factor of (\frac{1}{2}), what is the slope of line segment (bc)?

Explanation:

Step1: Identify Coordinates

From the graph, \( A(1,1) \), \( B(2,4) \), \( C(5,2) \).

Step2: Apply Dilation about A

Dilation about \( A(h,k) \) with scale factor \( s \): \( (x',y')=(h + s(x - h),k + s(y - k)) \).
For \( B \): \( x'=1+\frac{1}{2}(2 - 1)=1.5 \), \( y'=1+\frac{1}{2}(4 - 1)=2.5 \), so \( B'(1.5,2.5) \).
For \( C \): \( x'=1+\frac{1}{2}(5 - 1)=3 \), \( y'=1+\frac{1}{2}(2 - 1)=1.5 \), so \( C'(3,1.5) \).

Step3: Calculate Slope of \( B'C' \)

Slope formula: \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
\( m=\frac{1.5 - 2.5}{3 - 1.5}=\frac{-1}{1.5}=-\frac{2}{3} \)? Wait, no, wait: Wait, \( B(2,4) \), \( A(1,1) \): vector \( \overrightarrow{AB}=(1,3) \), after dilation \( \frac{1}{2} \), \( \overrightarrow{AB'}=\frac{1}{2}(1,3)=(0.5,1.5) \), so \( B'=A+\overrightarrow{AB'}=(1 + 0.5,1 + 1.5)=(1.5,2.5) \). \( C(5,2) \), vector \( \overrightarrow{AC}=(4,1) \), dilation: \( \overrightarrow{AC'}=\frac{1}{2}(4,1)=(2,0.5) \), so \( C'=A+\overrightarrow{AC'}=(1 + 2,1 + 0.5)=(3,1.5) \). Now slope \( B'C' \): \( \frac{1.5 - 2.5}{3 - 1.5}=\frac{-1}{1.5}=-\frac{2}{3} \)? Wait, no, wait original \( BC \): \( B(2,4) \), \( C(5,2) \), slope of \( BC \) is \( \frac{2 - 4}{5 - 2}=\frac{-2}{3} \). Dilation preserves slope (since it's a similarity transformation, parallel lines remain parallel, so slope of \( B'C' \) is same as slope of \( BC \)).
So slope of \( BC \): \( \frac{2 - 4}{5 - 2}=\frac{-2}{3} \), so slope of \( B'C' \) is also \( -\frac{2}{3} \)? Wait, no, wait my calculation for \( B' \) and \( C' \): Wait \( A(1,1) \), \( B(2,4) \): the vector from \( A \) to \( B \) is \( (2 - 1,4 - 1)=(1,3) \). Dilation by \( 1/2 \): new vector is \( (0.5,1.5) \), so \( B'=(1 + 0.5,1 + 1.5)=(1.5,2.5) \). \( C(5,2) \): vector from \( A \) to \( C \) is \( (5 - 1,2 - 1)=(4,1) \). Dilation by \( 1/2 \): new vector \( (2,0.5) \), so \( C'=(1 + 2,1 + 0.5)=(3,1.5) \). Now slope between \( (1.5,2.5) \) and \( (3,1.5) \): \( (1.5 - 2.5)/(3 - 1.5)=(-1)/1.5=-2/3 \). Alternatively, since dilation is a similarity transformation, \( B'C' \parallel BC \), so slope of \( B'C' \) equals slope of \( BC \). Slope of \( BC \): \( (2 - 4)/(5 - 2)=(-2)/3=-2/3 \). So the slope is \( -\frac{2}{3} \).

Answer:

\( -\dfrac{2}{3} \)