A triangle has corners at #(1, 7 )#, #( 8, 3 )#, and #( 4 , 8 )#. If the triangle is dilated by # 2 x# around #(2, 5)#, what will the new coordinates of its corners be?

Answer 1

#(0,9),(14,1)" and "(6,11)#

#"label the vertices of the triangle"#
#A(1,7),B(8,3)" and "C(4,8)#
#"and A', B', C' the images of A, B and C"#
#"let the centre of dilatation be "D(2,5)#
#rArrvec(DA')=color(red)(2)vec(DA)#
#rArrula'-uld=2(ula-uld)=2ula-2uld#
#rArrula'=2ula-uld#
#color(white)(rArrula')=2((1),(7))-((2),(5))=((0),(9))#
#"the coordinates of A' are the components of "ula'#
#rArrA'=(0,9)#
#"similarly "#
#vec(DB')=color(red)(2)vec(DB)#
#rArrulb'=2ulb-uld#
#color(white)(rArrulb')=2((8),(3))-((2),(5))=((14),(1))#
#rArrB'=(14,1)#
#"and "vec(DC')=color(red)(2)vec(DC)#
#rArrulc'=2ulc-uld#
#color(white)(rArrulc')=2((4),(8))-((2),(5))=((6),(11))#
#rArrC'=(6,11)#
#color(blue)"Result after dilatation"#
#(1,7)to(0,9),(8,3)to(14,1),(4,8)to(6,11)#
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Answer 2

To dilate the triangle by a factor of 2 around the point (2, 5), we'll use the dilation formula for coordinates (x', y'):

[ x' = k(x - h) + h ] [ y' = k(y - v) + v ]

Where:

  • (x, y) are the original coordinates of a point in the triangle,
  • (h, v) are the coordinates of the center of dilation (in this case, (2, 5)),
  • k is the scale factor of dilation (2 in this case),
  • (x', y') are the new coordinates after dilation.

Let's calculate the new coordinates for each corner of the triangle:

  1. For the point (1, 7): [ x' = 2(1 - 2) + 2 = 0 + 2 = 2 ] [ y' = 2(7 - 5) + 5 = 2 + 5 = 7 ] So, the new coordinates are (2, 7).

  2. For the point (8, 3): [ x' = 2(8 - 2) + 2 = 12 ] [ y' = 2(3 - 5) + 5 = 1 ] So, the new coordinates are (12, 1).

  3. For the point (4, 8): [ x' = 2(4 - 2) + 2 = 6 ] [ y' = 2(8 - 5) + 5 = 11 ] So, the new coordinates are (6, 11).

Therefore, after dilation by a factor of 2 around (2, 5), the new coordinates of the triangle's corners will be (2, 7), (12, 1), and (6, 11).

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