Question

draw each of the images described in the steps below. step 1: first, reflect the given triangle across the y - axis. step 2: then, using the resulting figure, perform a dilation centered at the point (1, 3) with a scale factor of 3.

Explanation:

Step 1: Identify Original Triangle Vertices

Assume the original triangle has vertices (let's find from the grid): Let's say the original triangle has vertices at, for example, \( A(-3, 4) \), \( B(-2, 4) \), \( C(-3, 5) \) (by looking at the grid, the triangle is near \( x=-3, -2 \) and \( y=4,5 \)).

Step 2: Reflect Across y - axis

The rule for reflection across the \( y \)-axis is \( (x,y)\to(-x,y) \).

• For \( A(-3, 4) \): New point \( A_1=(3, 4) \)
• For \( B(-2, 4) \): New point \( B_1=(2, 4) \)
• For \( C(-3, 5) \): New point \( C_1=(3, 5) \)

Step 3: Perform Dilation Centered at (1, 3) with Scale Factor 3

The formula for dilation centered at \( (h,k) \) with scale factor \( s \) is \( (x',y')=(h + s(x - h),k + s(y - k)) \)

• For \( A_1(3, 4) \):

\( x'=1+3(3 - 1)=1 + 6 = 7 \)
\( y'=3+3(4 - 3)=3 + 3 = 6 \)
So \( A_2=(7, 6) \)

• For \( B_1(2, 4) \):

\( x'=1+3(2 - 1)=1 + 3 = 4 \)
\( y'=3+3(4 - 3)=3 + 3 = 6 \)
So \( B_2=(4, 6) \)

• For \( C_1(3, 5) \):

\( x'=1+3(3 - 1)=1 + 6 = 7 \)
\( y'=3+3(5 - 3)=3 + 6 = 9 \)
So \( C_2=(7, 9) \)

Now, plot the reflected triangle with vertices \( (3,4),(2,4),(3,5) \) and then the dilated triangle with vertices \( (7,6),(4,6),(7,9) \) on the grid.

Answer:

To draw the final figure:

1. Reflect the original triangle across the \( y \)-axis to get a triangle with vertices (3, 4), (2, 4), (3, 5).
2. Dilate this reflected triangle with center (1, 3) and scale factor 3 to get a triangle with vertices (7, 6), (4, 6), (7, 9). Plot these points and connect them to form the final triangle. (Note: Since this is a drawing - based problem, the key is to follow the transformation steps for the vertices as calculated above and plot them on the given grid.)