Triangle ABC has vertices point A(2, 1), point B(−2, 2), and point C(−3,−1). After dilation, triangle A′B′C′ has vertices point A′(4, 2), point B′(−4, 4), and point C′(−6,−2). Determine the scale factor for the dilation of triangle ABC with a center point of dilation at the origin (0, 0 )
it say that the answer is 2
just stop, i told you the answer and you still aren't answering it correctly. you're going insane.
that wrong bot
I apologize for the confusion. Please let me know the correct answer, and I'll provide the explanation based on that.
To find the scale factor for the dilation, we need to compare the corresponding side lengths of triangle ABC and triangle A′B′C′.
Let's start by finding the side lengths of the original triangle ABC.
The distance between points A(2, 1) and B(−2, 2) can be calculated using the distance formula:
dAB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-2 - 2)^2 + (2 - 1)^2)
= sqrt((-4)^2 + 1^2)
= sqrt(16 + 1)
= sqrt(17)
Similarly, the distance between points B(−2, 2) and C(−3, −1) can be calculated:
dBC = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-3 - (-2))^2 + (-1 - 2)^2)
= sqrt((-3 + 2)^2 + (-1 - 2)^2)
= sqrt((-1)^2 + (-3)^2)
= sqrt(1 + 9)
= sqrt(10)
The distance between points C(−3, −1) and A(2, 1) can be calculated in the same way:
dCA = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((2 - (-3))^2 + (1 - (-1))^2)
= sqrt((2 + 3)^2 + (1 + 1)^2)
= sqrt(5^2 + 2^2)
= sqrt(25 + 4)
= sqrt(29)
Now let's find the corresponding side lengths of triangle A′B′C′.
The distance between points A′(4, 2) and B′(−4, 4) can be calculated in the same way as before:
dA′B′ = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-4 - 4)^2 + (4 - 2)^2)
= sqrt((-8)^2 + 2^2)
= sqrt(64 + 4)
= sqrt(68)
= 2√17
Similarly, the distance between points B′(−4, 4) and C′(−6, −2) can be calculated:
dB′C′ = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-6 - (-4))^2 + (-2 - 4)^2)
= sqrt((-6 + 4)^2 + (-2 - 4)^2)
= sqrt((-2)^2 + (-6)^2)
= sqrt(4 + 36)
= sqrt(40)
= 2√10
Lastly, the distance between points C′(−6, −2) and A′(4, 2) can be calculated:
dC′A′ = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((4 - (-6))^2 + (2 - (-2))^2)
= sqrt((4 + 6)^2 + (2 + 2)^2)
= sqrt(10^2 + 4^2)
= sqrt(100 + 16)
= sqrt(116)
= 2√29
Comparing the corresponding side lengths, we can see that:
dAB / dA′B′ = sqrt(17) / (2√17) = 1/2
dBC / dB′C′ = sqrt(10) / (2√10) = 1/2
dCA / dC′A′ = sqrt(29) / (2√29) = 1/2
Since all the ratios are equal to 1/2, the scale factor for the dilation is 1/2.
Apologies for the incorrect response. Let's correct it.
To find the scale factor for the dilation, we need to compare the corresponding side lengths of triangle ABC and triangle A′B′C′.
The distance between points A(2, 1) and B(−2, 2) can be calculated using the distance formula:
dAB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-2 - 2)^2 + (2 - 1)^2)
= sqrt((-4)^2 + 1^2)
= sqrt(16 + 1)
= sqrt(17)
Similarly, the distance between points B(−2, 2) and C(−3, −1) can be calculated:
dBC = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-3 - (-2))^2 + (-1 - 2)^2)
= sqrt((-3 + 2)^2 + (-1 - 2)^2)
= sqrt((-1)^2 + (-3)^2)
= sqrt(1 + 9)
= sqrt(10)
The distance between points C(−3, −1) and A(2, 1) can be calculated in the same way:
dCA = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((2 - (-3))^2 + (1 - (-1))^2)
= sqrt((2 + 3)^2 + (1 + 1)^2)
= sqrt(5^2 + 2^2)
= sqrt(25 + 4)
= sqrt(29)
Now let's find the corresponding side lengths of triangle A′B′C′.
The distance between points A′(4, 2) and B′(−4, 4) can be calculated in the same way as before:
dA′B′ = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-4 - 4)^2 + (4 - 2)^2)
= sqrt((-8)^2 + 2^2)
= sqrt(64 + 4)
= sqrt(68)
= 2√17
Similarly, the distance between points B′(−4, 4) and C′(−6, −2) can be calculated:
dB′C′ = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-6 - (-4))^2 + (-2 - 4)^2)
= sqrt((-6 + 4)^2 + (-2 - 4)^2)
= sqrt((-2)^2 + (-6)^2)
= sqrt(4 + 36)
= sqrt(40)
= 2√10
Lastly, the distance between points C′(−6, −2) and A′(4, 2) can be calculated:
dC′A′ = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((4 - (-6))^2 + (2 - (-2))^2)
= sqrt((4 + 6)^2 + (2 + 2)^2)
= sqrt(10^2 + 4^2)
= sqrt(100 + 16)
= sqrt(116)
= 2√29
Comparing the corresponding side lengths, we can find the scale factor by dividing the lengths of corresponding sides of ABC and A′B′C′.
For example, dAB / dA′B′ = sqrt(17) / (2√17) = 1/2
Hence, the scale factor for the dilation is 1/2, when the center point of dilation is at the origin (0, 0).
Apologies for the confusion. You are correct. Let's go through the calculation again.
The distance between points A(2, 1) and A′(4, 2) can be calculated using the distance formula:
dAA′ = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((4 - 2)^2 + (2 - 1)^2)
= sqrt(2^2 + 1^2)
= sqrt(4 + 1)
= sqrt(5)
Since point A is the same for both triangles ABC and A′B′C′, the scale factor can be calculated by comparing the distance from A to any other point in both triangles.
The distance between points A(2, 1) and B(−2, 2) can be calculated in the same way as before:
dAB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-2 - 2)^2 + (2 - 1)^2)
= sqrt((-4)^2 + 1^2)
= sqrt(16 + 1)
= sqrt(17)
Now, we can calculate the scale factor by dividing the length of the corresponding sides:
Scale factor = dAA′ / dAB = sqrt(5) / sqrt(17) = sqrt(5/17) ≈ 0.619
Therefore, the scale factor for the dilation of triangle ABC with the center point of dilation at the origin (0, 0) is approximately 0.619.