Questions: Dilate Figure QRSTUV with center of dilation at Point R by a scale factor of 1/2.

Dilate Figure QRSTUV with center of dilation at Point R by a scale factor of 1/2.
Transcript text: Dilate Figure QRSTUV with center of dilation at Point $R$ by a scale factor of $\frac{1}{2}$.
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Solution

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Solution Steps

Step 1: Identify the coordinates of each vertex

The coordinates of the vertices of the hexagon QRSTUV are: Q(-2, 2) R(1, 8) S(6, 8) T(10, 2) U(7, -4) V(2, -4)

Step 2: Find the vector from the center of dilation (R) to each vertex

We need to find the vector from R(1, 8) to each other vertex. This is done by subtracting the coordinates of R from the coordinates of each vertex. RQ = Q - R = (-2 - 1, 2 - 8) = (-3, -6) RS = S - R = (6 - 1, 8 - 8) = (5, 0) RT = T - R = (10 - 1, 2 - 8) = (9, -6) RU = U - R = (7 - 1, -4 - 8) = (6, -12) RV = V - R = (2 - 1, -4 - 8) = (1, -12)

Step 3: Multiply each vector by the scale factor

The scale factor is $\frac{1}{2}$. We multiply each component of the vectors by $\frac{1}{2}$. RQ' = $\frac{1}{2}$RQ = $\frac{1}{2}$(-3, -6) = (-1.5, -3) RS' = $\frac{1}{2}$RS = $\frac{1}{2}$(5, 0) = (2.5, 0) RT' = $\frac{1}{2}$RT = $\frac{1}{2}$(9, -6) = (4.5, -3) RU' = $\frac{1}{2}$RU = $\frac{1}{2}$(6, -12) = (3, -6) RV' = $\frac{1}{2}$RV = $\frac{1}{2}$(1, -12) = (0.5, -6)

Step 4: Add the resulting vectors to the center of dilation

We add the scaled vectors to the coordinates of R(1, 8) to get the coordinates of the dilated image. Q' = R + RQ' = (1 - 1.5, 8 - 3) = (-0.5, 5) S' = R + RS' = (1 + 2.5, 8 + 0) = (3.5, 8) T' = R + RT' = (1 + 4.5, 8 - 3) = (5.5, 5) U' = R + RU' = (1 + 3, 8 - 6) = (4, 2) V' = R + RV' = (1 + 0.5, 8 - 6) = (1.5, 2) R' = R = (1, 8) since R is the center of dilation

Final Answer

The coordinates of the dilated image Q'R'S'T'U'V' are Q'(-0.5, 5), R'(1, 8), S'(3.5, 8), T'(5.5, 5), U'(4, 2), and V'(1.5, 2). \( \boxed{Q'(-0.5, 5), R'(1, 8), S'(3.5, 8), T'(5.5, 5), U'(4, 2), V'(1.5, 2)} \)

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