A triangle has corners at #(6 ,9 )#, #(-2 ,-1 )#, and #(1 ,-1 )#. If the triangle is dilated by a factor of #1/3 # about point #(-5 ,-2 ), how far will its centroid move?
By signing up, you agree to our Terms of Service and Privacy Policy
To find the centroid of a triangle, you take the average of the x-coordinates of the vertices to find the x-coordinate of the centroid, and similarly for the y-coordinate.
The centroid of a triangle is given by the coordinates (x_c, y_c), where: [ x_c = \frac{x_1 + x_2 + x_3}{3} ] [ y_c = \frac{y_1 + y_2 + y_3}{3} ]
Given the coordinates of the vertices: [ (6, 9), (-2, -1), (1, -1) ]
We can substitute these values into the formulas to find the centroid of the original triangle.
The x-coordinate of the centroid: [ x_c = \frac{6 + (-2) + 1}{3} = \frac{5}{3} ]
The y-coordinate of the centroid: [ y_c = \frac{9 + (-1) + (-1)}{3} = \frac{7}{3} ]
Now, if we dilate the triangle by a factor of ( \frac{1}{3} ) about the point (-5, -2), we need to find how far the centroid moves.
The distance between the original centroid and (-5, -2) is given by the distance formula: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
Substituting the coordinates: [ d = \sqrt{(\frac{5}{3} - (-5))^2 + (\frac{7}{3} - (-2))^2} ]
[ d = \sqrt{(\frac{5}{3} + 5)^2 + (\frac{7}{3} + 2)^2} ]
[ d = \sqrt{(\frac{20}{3})^2 + (\frac{13}{3})^2} ]
[ d = \sqrt{\frac{400}{9} + \frac{169}{9}} ]
[ d = \sqrt{\frac{569}{9}} ]
So the distance the centroid moves is ( \frac{\sqrt{569}}{3} ).
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
- How do you locate the center of a dilation?
- What are some examples of dilation?
- A line segment has endpoints at #(3 ,5 )# and #(6 ,1 )#. The line segment is dilated by a factor of #4 # around #(6 ,2 )#. What are the new endpoints and length of the line segment?
- Points A and B are at #(3 ,7 )# and #(4 ,2 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #5 #. If point A is now at point B, what are the coordinates of point C?
- A triangle has corners at #(-2 ,6 )#, #(4 ,-3 )#, and #(2 ,9 )#. If the triangle is dilated by a factor of #5 # about point #(-7 ,1 ), how far will its centroid move?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7