# A triangle has corners at #(1, 5 )#, ( 6, 8)#, and #( 4, 2)#. If the triangle is reflected across the x-axis, what will its new centroid be?

Coordinates of new centroid

Centroid coordinates

Since the triangle is reflected across x-axis, x coordinate of new centroid will remain the same. Only y coordinate will mirror and equidistant from the x-axis.

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To find the centroid of a triangle after reflecting it across the x-axis, we use the property that the centroid of a triangle remains the same after reflection across the x-axis.

The centroid of a triangle is the point of intersection of its medians. A median is a line segment joining a vertex of a triangle to the midpoint of the opposite side.

Let's label the given points as A(1, 5), B(6, 8), and C(4, 2).

The midpoint of side AB is ((1 + 6)/2, (5 + 8)/2) = (3.5, 6.5). The midpoint of side BC is ((6 + 4)/2, (8 + 2)/2) = (5, 5). The midpoint of side AC is ((1 + 4)/2, (5 + 2)/2) = (2.5, 3.5).

The centroid of the triangle ABC is the point of intersection of the medians, which are the lines connecting each vertex to the midpoint of the opposite side.

So, the centroid of triangle ABC is the average of the coordinates of the midpoints:

Centroid = ((3.5 + 5 + 2.5)/3, (6.5 + 5 + 3.5)/3) = ((11/3), (15/3)) = (11/3, 5).

Since reflecting the triangle across the x-axis doesn't change its centroid, the new centroid remains at the same point: (11/3, 5).

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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.

- A triangle has corners at #(3, 4 )#, ( 4, -2)#, and #( 7, -1)#. If the triangle is reflected across the x-axis, what will its new centroid be?
- Points A and B are at #(8 ,5 )# and #(8 ,2 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #3 #. If point A is now at point B, what are the coordinates of point C?
- Points A and B are at #(1 ,8 )# and #(3 ,6 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #2 #. If point A is now at point B, what are the coordinates of point C?
- Circle A has a radius of #2 # and a center of #(3 ,1 )#. Circle B has a radius of #4 # and a center of #(8 ,5 )#. If circle B is translated by #<-4 ,-1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A line segment has endpoints at #(1 ,2 )# and #(9 ,4 )#. If the line segment is rotated about the origin by #( 3 pi)/2 #, translated vertically by # 3 #, and reflected about the x-axis, what will the line segment's new endpoints be?

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