A triangle has corners at #(5, 8 )#, #( 3, 1 )#, and #( 9 , 2 )#. If the triangle is dilated by # 3 x# around #(4, 2)#, what will the new coordinates of its corners be?
New coordinates are
Dilated around D(4,2) by a factor of 3.
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To dilate a triangle by a factor of (3) around the point ((4, 2)), you apply the dilation formula to each corner of the triangle. The formula for dilation from a point ((x_0, y_0)) by a scale factor (k) is given by:
[ \begin{align*} x' &= x_0 + k(x - x_0) \ y' &= y_0 + k(y - y_0) \end{align*} ]
where ((x, y)) are the original coordinates, and ((x', y')) are the dilated coordinates.
Applying this formula to each corner of the triangle:
- For the corner at ((5, 8)):
[ \begin{align*} x' &= 4 + 3(5 - 4) = 4 + 3(1) = 4 + 3 = 7 \ y' &= 2 + 3(8 - 2) = 2 + 3(6) = 2 + 18 = 20 \end{align*} ]
So, the new coordinates are ((7, 20)).
- For the corner at ((3, 1)):
[ \begin{align*} x' &= 4 + 3(3 - 4) = 4 - 3 = 1 \ y' &= 2 + 3(1 - 2) = 2 - 3 = -1 \end{align*} ]
So, the new coordinates are ((1, -1)).
- For the corner at ((9, 2)):
[ \begin{align*} x' &= 4 + 3(9 - 4) = 4 + 3(5) = 4 + 15 = 19 \ y' &= 2 + 3(2 - 2) = 2 + 0 = 2 \end{align*} ]
So, the new coordinates are ((19, 2)).
Therefore, after the dilation by a factor of (3) around the point ((4, 2)), the new coordinates of the triangle's corners are ((7, 20)), ((1, -1)), and ((19, 2)).
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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.
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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- How do dilations relate to similarity?
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