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Sagot :
To find the correct statements about the dilated image of triangle QRS, we will follow these steps:
1. Dilate each point using the rule Do,2 (x,y) = (2x, 2y):
- For point Q(-3, 3):
Q' will have coordinates (2(-3), 23) = (-6, 6).
- For point S(-1, 1):
S' will have coordinates (2(-1), 21) = (-2, 2).
- For point R(2, 4):
R' will have coordinates (22, 24) = (4, 8).
So, the coordinates of the dilated points are:
- Q'(-6, 6)
- S'(-2, 2)
- R'(4, 8)
2. Calculate the slope of side Q'S':
The slope between two points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1).
For points Q'(-6, 6) and S'(-2, 2):
[tex]\( \text{Slope} = \frac{2 - 6}{-2 - (-6)} = \frac{-4}{4} = -1 \)[/tex].
3. Compare the lengths of QR and Q'R':
The length of a segment between two points (x1, y1) and (x2, y2) is given by:
[tex]\( \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \)[/tex].
For QR, where Q = (-3, 3) and R = (2, 4):
[tex]\[ \text{Length of QR} = \sqrt{(2 - (-3))^2 + (4 - 3)^2} = \sqrt{(2 + 3)^2 + 1^2} = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26} \approx 5.099 \][/tex]
For Q'R', where Q' = (-6, 6) and R' = (4, 8):
[tex]\[ \text{Length of Q'R'} = \sqrt{(4 - (-6))^2 + (8 - 6)^2} = \sqrt{(4 + 6)^2 + 2^2} = \sqrt{10^2 + 2^2} = \sqrt{100 + 4} = \sqrt{104} \approx 10.198 \][/tex]
4. Compare the distance from Q and Q' to the origin:
The distance from a point (x, y) to the origin (0, 0) is given by:
[tex]\( \sqrt{x^2 + y^2} \)[/tex].
For Q(-3, 3):
[tex]\[ \text{Distance from Q to origin} = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.243 \][/tex]
For Q'(-6, 6):
[tex]\[ \text{Distance from Q' to origin} = \sqrt{(-6)^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} \approx 8.485 \][/tex]
Notice that the distance from Q' to the origin is indeed twice the distance from Q to the origin.
Based on the above calculations, the correct statements are:
1. Side Q'S' lies on a line with a slope of -1.
2. QR is shorter (not longer) than Q'R'.
3. The distance from Q' to the origin is twice the distance from Q to the origin.
Therefore, the three true statements are:
- Side Q'S' lies on a line with a slope of -1.
- QR is shorter, not longer, than Q'R'.
- The distance from Q' to the origin is twice the distance from Q to the origin.
1. Dilate each point using the rule Do,2 (x,y) = (2x, 2y):
- For point Q(-3, 3):
Q' will have coordinates (2(-3), 23) = (-6, 6).
- For point S(-1, 1):
S' will have coordinates (2(-1), 21) = (-2, 2).
- For point R(2, 4):
R' will have coordinates (22, 24) = (4, 8).
So, the coordinates of the dilated points are:
- Q'(-6, 6)
- S'(-2, 2)
- R'(4, 8)
2. Calculate the slope of side Q'S':
The slope between two points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1).
For points Q'(-6, 6) and S'(-2, 2):
[tex]\( \text{Slope} = \frac{2 - 6}{-2 - (-6)} = \frac{-4}{4} = -1 \)[/tex].
3. Compare the lengths of QR and Q'R':
The length of a segment between two points (x1, y1) and (x2, y2) is given by:
[tex]\( \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \)[/tex].
For QR, where Q = (-3, 3) and R = (2, 4):
[tex]\[ \text{Length of QR} = \sqrt{(2 - (-3))^2 + (4 - 3)^2} = \sqrt{(2 + 3)^2 + 1^2} = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26} \approx 5.099 \][/tex]
For Q'R', where Q' = (-6, 6) and R' = (4, 8):
[tex]\[ \text{Length of Q'R'} = \sqrt{(4 - (-6))^2 + (8 - 6)^2} = \sqrt{(4 + 6)^2 + 2^2} = \sqrt{10^2 + 2^2} = \sqrt{100 + 4} = \sqrt{104} \approx 10.198 \][/tex]
4. Compare the distance from Q and Q' to the origin:
The distance from a point (x, y) to the origin (0, 0) is given by:
[tex]\( \sqrt{x^2 + y^2} \)[/tex].
For Q(-3, 3):
[tex]\[ \text{Distance from Q to origin} = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.243 \][/tex]
For Q'(-6, 6):
[tex]\[ \text{Distance from Q' to origin} = \sqrt{(-6)^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} \approx 8.485 \][/tex]
Notice that the distance from Q' to the origin is indeed twice the distance from Q to the origin.
Based on the above calculations, the correct statements are:
1. Side Q'S' lies on a line with a slope of -1.
2. QR is shorter (not longer) than Q'R'.
3. The distance from Q' to the origin is twice the distance from Q to the origin.
Therefore, the three true statements are:
- Side Q'S' lies on a line with a slope of -1.
- QR is shorter, not longer, than Q'R'.
- The distance from Q' to the origin is twice the distance from Q to the origin.
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