Question

abc is shown. triangle abc is rotated 90° counterclockwise about point a and then translated along <2,3>. what is the location of the image of point b?

Explanation:

Step1: Determine the coordinates of point B

From the graph, point \( A=(1,1) \), point \( B=(3,2) \).

Step2: Rotate point B \( 90^{\circ} \) counter - clockwise about point A

The formula for rotating a point \( (x,y) \) \( 90^{\circ} \) counter - clockwise about a point \( (a,b) \) is \( (x',y')=(a-(y - b),b+(x - a)) \).
Substitute \( a = 1,b = 1,x = 3,y = 2 \) into the formula:
\( x'=1-(2 - 1)=0 \), \( y'=1+(3 - 1)=3 \). So the rotated point \( B_1=(0,3) \).

Step3: Translate the rotated point

The translation vector is \( \langle2,3
angle \). The formula for translation is \( (x'',y'')=(x'+h,y'+k) \), where \( h = 2,k = 3 \).
Substitute \( x' = 0,y' = 3 \) into the formula: \( x''=0 + 2=2 \), \( y''=3+3 = 6 \).

Answer:

\((2,6)\)