Question

the rectangle qrst is a dilation of the rectangle qrst. what is the scale factor of the dilation? simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Find side lengths of original rectangle

For rectangle QRST, let's find length of TQ (horizontal side). T is at (-8,2), Q is at (10,2). So length \( TQ = 10 - (-8)=18 \)? Wait, no, wait. Wait, looking at the grid, T is at (-8,2), Q is at (10,2)? Wait no, maybe I misread. Wait, T is at (-8,2), Q is at (10,2)? Wait, no, the horizontal distance: from x=-8 to x=10, that's 18? Wait, no, maybe the vertical side. Wait, S is at (-8,-10), R is at (10,-10)? Wait no, the green rectangle: T(-8,2), Q(10,2), R(10,-10), S(-8,-10). So length of TQ: 10 - (-8) = 18? Wait, no, 10 - (-8) is 18? Wait, 10 - (-8) = 18? Wait, 10 +8=18. Then the vertical side: from y=2 to y=-10, that's 12 units? Wait, no, 2 - (-10)=12. Now the yellow rectangle Q'R'S'T': T'(-4,1), Q'(5,1), R'(5,-5), S'(-4,-5). Wait, let's check coordinates. T' is at (-4,1), Q' at (5,1). So length T'Q' = 5 - (-4)=9. Then vertical side: 1 - (-5)=6. Now, scale factor is (length of image)/(length of original). So for horizontal side: 9/18 = 1/2. For vertical side: 6/12 = 1/2. So scale factor is 1/2.
Wait, maybe better to take a side. Let's take T to T': original T is (-8,2), image T' is (-4,1). Wait, no, maybe the horizontal side of the top. Original top side: from x=-8 to x=10, length 18? Wait, no, wait the x-coordinates: T is (-8,2), Q is (10,2). So the horizontal length is 10 - (-8) = 18. The image Q'R'S'T': T' is (-4,1), Q' is (5,1). So horizontal length is 5 - (-4) = 9. So scale factor is 9/18 = 1/2. Alternatively, vertical side: original vertical side (from y=2 to y=-10) is 12 units (2 - (-10)=12). Image vertical side: from y=1 to y=-5, that's 6 units (1 - (-5)=6). 6/12=1/2. So scale factor is 1/2.

Step2: Confirm with another side

Take the horizontal side: original length (TQ) is 18 (from x=-8 to x=10), image length (T'Q') is 9 (from x=-4 to x=5). 9/18 = 1/2. Vertical side: original (TS) from y=2 to y=-10 is 12, image (T'S') from y=1 to y=-5 is 6. 6/12=1/2. So scale factor is 1/2.

Answer:

\(\frac{1}{2}\)