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Math  /  Geometry

QuestionDraw the image of the quadrilateral under a dilation about point AA with a scale factor of 13\frac{1}{3}.
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Studdy Solution

STEP 1

1. A dilation transformation changes the size of a figure by a scale factor relative to a fixed point called the center of dilation.
2. The center of dilation in this problem is point AA with coordinates (5,3)(5, 3).
3. The scale factor for the dilation is given as 13\frac{1}{3}.
4. The vertices of the original quadrilateral are at the points (−4,−2)(-4, -2), (2,−2)(2, -2), (4,6)(4, 6), and (12,6)(12, 6).
5. The coordinates of the dilated quadrilateral can be found by applying the dilation formula to each vertex of the original quadrilateral.

STEP 2

1. Identify the coordinates of the vertices of the original quadrilateral.
2. Apply the dilation formula to each vertex.
3. Calculate the coordinates of the vertices of the dilated quadrilateral.
4. Verify the coordinates of the new vertices by comparing them with the image provided.

STEP 3

Identify the coordinates of the vertices of the original quadrilateral.
The vertices are: A=(5,3) A = (5, 3) B=(−4,−2) B = (-4, -2) C=(2,−2) C = (2, -2) D=(4,6) D = (4, 6) E=(12,6) E = (12, 6)

STEP 4

Apply the dilation formula to each vertex.
The dilation formula for a point (x,y)(x, y) with center of dilation (x0,y0)(x_0, y_0) and scale factor kk is: (x′,y′)=(x0+k(x−x0),y0+k(y−y0)) (x', y') = \left( x_0 + k(x - x_0), y_0 + k(y - y_0) \right)

STEP 5

Calculate the dilated coordinates for point B=(−4,−2)B = (-4, -2).
B′=(5+13(−4−5),3+13(−2−3))=(5+13(−9),3+13(−5))=(5−3,3−53)=(2,43) B' = \left( 5 + \frac{1}{3}(-4 - 5), 3 + \frac{1}{3}(-2 - 3) \right) = \left( 5 + \frac{1}{3}(-9), 3 + \frac{1}{3}(-5) \right) = \left( 5 - 3, 3 - \frac{5}{3} \right) = \left( 2, \frac{4}{3} \right)

STEP 6

Calculate the dilated coordinates for point C=(2,−2)C = (2, -2).
C′=(5+13(2−5),3+13(−2−3))=(5+13(−3),3+13(−5))=(5−1,3−53)=(4,43) C' = \left( 5 + \frac{1}{3}(2 - 5), 3 + \frac{1}{3}(-2 - 3) \right) = \left( 5 + \frac{1}{3}(-3), 3 + \frac{1}{3}(-5) \right) = \left( 5 - 1, 3 - \frac{5}{3} \right) = \left( 4, \frac{4}{3} \right)

STEP 7

Calculate the dilated coordinates for point D=(4,6)D = (4, 6).
D′=(5+13(4−5),3+13(6−3))=(5+13(−1),3+13(3))=(5−13,3+1)=(143,4) D' = \left( 5 + \frac{1}{3}(4 - 5), 3 + \frac{1}{3}(6 - 3) \right) = \left( 5 + \frac{1}{3}(-1), 3 + \frac{1}{3}(3) \right) = \left( 5 - \frac{1}{3}, 3 + 1 \right) = \left( \frac{14}{3}, 4 \right)

STEP 8

Calculate the dilated coordinates for point E=(12,6)E = (12, 6).
E′=(5+13(12−5),3+13(6−3))=(5+13(7),3+13(3))=(5+73,3+1)=(223,4) E' = \left( 5 + \frac{1}{3}(12 - 5), 3 + \frac{1}{3}(6 - 3) \right) = \left( 5 + \frac{1}{3}(7), 3 + \frac{1}{3}(3) \right) = \left( 5 + \frac{7}{3}, 3 + 1 \right) = \left( \frac{22}{3}, 4 \right)

STEP 9

Verify the coordinates of the dilated vertices by comparing them with the image provided.
The vertices of the dilated quadrilateral are: B′=(2,43) B' = \left( 2, \frac{4}{3} \right) C′=(4,43) C' = \left( 4, \frac{4}{3} \right) D′=(143,4) D' = \left( \frac{14}{3}, 4 \right) E′=(223,4) E' = \left( \frac{22}{3}, 4 \right)
These coordinates match the approximate locations given in the image.

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