A triangle has corners at #(3 ,1 )#, #(1 ,-2 )#, and #(-4 ,-5 )#. If the triangle is dilated by a factor of #5 # about point #(7 ,-6 ), how far will its centroid move?

Answer 1

# 4 sqrt{65} #

The centroid is the center of mass of the triangle and is given by the average of the coordinates:

# C=\frac 1 3(3+1-4,1-2-5)=(0,-2)#
The distance from the centroid to the dilation point #(7,-6)# is
#d = \sqrt{(0-7)^2+(-2 - -6)^2} = sqrt{65} #
The factor of five dilation means this will end up #5 sqrt{65}# from the dilation point after dilation, so will have moved #4 sqrt{65}#.
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Answer 2

To find the centroid of the triangle before and after dilation, follow these steps:

  1. Find the coordinates of the centroid of the original triangle using the formula: [ \text{Centroid} = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) ]

  2. Dilate the coordinates of the centroid using the given dilation factor and center.

  3. Calculate the displacement of the centroid by finding the difference between the dilated centroid and the original centroid.

Let's calculate:

Original triangle coordinates: ( A(3, 1) ) ( B(1, -2) ) ( C(-4, -5) )

  1. Calculate the coordinates of the centroid of the original triangle: [ \text{Centroid} = \left( \frac{3 + 1 - 4}{3}, \frac{1 - 2 - 5}{3} \right) = \left( \frac{0}{3}, \frac{-6}{3} \right) = (0, -2) ]

  2. Dilate the coordinates of the centroid: Dilation factor: 5 Center of dilation: ( D(7, -6) )

[ \text{Dilated Centroid} = (7 + 5 \times 0, -6 + 5 \times (-2)) = (7, -16) ]

  1. Calculate the displacement of the centroid: [ \text{Displacement} = \text{Dilated Centroid} - \text{Original Centroid} ] [ \text{Displacement} = (7, -16) - (0, -2) = (7, -16) - (0, -2) = (7, -14) ]

So, the centroid will move 7 units to the right and 14 units upward.

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Answer from HIX Tutor

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.

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