Question

triangle abc has vertices point a (2,1), point b (4, 3), and point c (-2, 5). find the coordinates
Answer
triangle abc has vertices point a (2,1), point b (4, 3), and point c (-2, 5). find the coordinates a', b', and c' after a dilation with a scale factor of 7 and a center point of dilation at the origin (0,0)

a' = (_,_)
b' = (_,_)
c' = (_,_)
Answer
if AB = 27 inches, then find the distance of A'B' after a reflection over the x-axis and a translation of (x + 6, y - 2)

A'B' = ___
Answer
thats wrong
Answer
thats still wrong
Answer
what is point c (2, 2) rotated 305 degrees clockwise
Answered by GPT 3.5
of the centroid of triangle ABC.

To find the coordinates of the centroid of a triangle, we need to find the average of the x-coordinates and the average of the y-coordinates of its vertices.

The x-coordinate of the centroid is found by adding up the x-coordinates of A, B, and C, and then dividing the sum by 3:

x-coordinate of centroid = (2 + 4 + (-2))/3 = 4/3

The y-coordinate of the centroid is found by adding up the y-coordinates of A, B, and C, and then dividing the sum by 3:

y-coordinate of centroid = (1 + 3 + 5)/3 = 3

Therefore, the coordinates of the centroid of triangle ABC are (4/3, 3).
Answered by GPT 3.5
To find the coordinates after a dilation with a scale factor of 7 and a center point of dilation at the origin (0,0), we need to multiply the x and y coordinates of each vertex by the scale factor.

For point A (2, 1):

x coordinate of A' = 2 * 7 = 14
y coordinate of A' = 1 * 7 = 7

So, A' = (14, 7)

For point B (4, 3):

x coordinate of B' = 4 * 7 = 28
y coordinate of B' = 3 * 7 = 21

So, B' = (28, 21)

For point C (-2, 5):

x coordinate of C' = -2 * 7 = -14
y coordinate of C' = 5 * 7 = 35

So, C' = (-14, 35)

Therefore, the coordinates after the dilation are:
a' = (14, 7)
b' = (28, 21)
c' = (-14, 35)
Answered by GPT 3.5
To find the distance of A'B' after a reflection over the x-axis and a translation of (x + 6, y - 2), we first need to reflect the coordinates of A' and B' over the x-axis.

For A' (14, 7):

Reflecting over the x-axis changes the sign of the y-coordinate:

A" = (14, -7)

For B' (28, 21):

Reflecting over the x-axis changes the sign of the y-coordinate:

B" = (28, -21)

Next, we need to apply the translation of (x + 6, y - 2) to the coordinates A" and B":

For A" (14, -7):

x coordinate of A"' = 14 + 6 = 20
y coordinate of A"' = -7 - 2 = -9

So, A"' = (20, -9)

For B" (28, -21):

x coordinate of B"' = 28 + 6 = 34
y coordinate of B"' = -21 - 2 = -23

So, B"' = (34, -23)

Now, we can find the distance between A"' and B"' using the distance formula:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Distance = sqrt((34 - 20)^2 + (-23 - (-9))^2)

Distance = sqrt((14)^2 + (-14)^2)

Distance = sqrt(196 + 196)

Distance = sqrt(392)

Distance ≈ 19.80 inches

Therefore, the distance of A'B' after the reflection and translation is approximately 19.80 inches.
Answered by GPT 3.5
Apologies for the error in my previous response. Let's re-calculate the distance of A'B' after the reflection over the x-axis and the translation of (x + 6, y - 2).

For A' (14, 7), after reflecting over the x-axis, the y-coordinate becomes negative:

A" = (14, -7)

Then, applying the translation of (x + 6, y - 2):

A"' = (14 + 6, -7 - 2) = (20, -9)

For B' (28, 21), after reflecting over the x-axis:

B" = (28, -21)

Then, applying the translation:

B"' = (28 + 6, -21 - 2) = (34, -23)

Now, let's calculate the distance between A"' (20, -9) and B"' (34, -23):

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Distance = sqrt((34 - 20)^2 + (-23 - (-9))^2)

Distance = sqrt((14)^2 + (-14)^2)

Distance = sqrt(196 + 196)

Distance = sqrt(392)

Distance ≈ 19.80

Therefore, the distance of A'B' after the reflection over the x-axis and the translation of (x + 6, y - 2) is approximately 19.80 units (since the units were not specified in the question).
Answered by GPT 3.5
Apologies once again for the mistake. Let's recalculate the distance of A'B' after the reflection over the x-axis and the translation of (x + 6, y - 2).

For A' (14, 7), after reflecting over the x-axis, the y-coordinate becomes negative:

A" = (14, -7)

Then, applying the translation of (x + 6, y - 2):

A"' = (14 + 6, -7 - 2) = (20, -9)

For B' (28, 21), after reflecting over the x-axis:

B" = (28, -21)

Then, applying the translation:

B"' = (28 + 6, -21 - 2) = (34, -23)

Now, let's calculate the distance between A"' (20, -9) and B"' (34, -23):

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Distance = sqrt((34 - 20)^2 + (-23 - (-9))^2)

Distance = sqrt((14)^2 + (-14)^2)

Distance = sqrt(196 + 196)

Distance = sqrt(392)

Distance = sqrt(392) ≈ 19.80

Therefore, the distance of A'B' after the reflection over the x-axis and the translation of (x + 6, y - 2) is approximately 19.80 units (since the units were not specified in the question).
Answered by GPT 3.5
To rotate point C (2, 2) 305 degrees clockwise, we can use the rotation formula:

x' = x * cos(theta) - y * sin(theta)
y' = x * sin(theta) + y * cos(theta)

where (x, y) are the original coordinates of the point, and theta is the angle of rotation.

Converting 305 degrees to radians: 305 degrees * (pi / 180 radians/degree) = 5.34 radians (rounded to 2 decimal places).

Using the rotation formula:

x' = 2 * cos(5.34) - 2 * sin(5.34)
y' = 2 * sin(5.34) + 2 * cos(5.34)

Evaluating the trigonometric functions:

x' ≈ 2 * -0.77 - 2 * 0.63
≈ -1.54 - 1.26
≈ -2.80

y' ≈ 2 * 0.63 + 2 * -0.77
≈ 1.26 - 1.54
≈ -0.28

Therefore, point C (2, 2) rotated 305 degrees clockwise is approximately (-2.80, -0.28).