A triangle has corners at #(2, 1 )#, #( 1, 3 )#, and #(5 , 5 )#. If the triangle is dilated by # 3 x# around #(1, 6)#, what will the new coordinates of its corners be?

2 Answers
Douglas K.
May 10, 2017

#(2,1)to(4,-9)#
#(1,3)to(1,-3)#
#(5,5)to(13,3)#

Explanation:

The vector equation of a line from #(1, 6)# to #(2,1)# is:

#(x,y)=(1,6)+t((2-1)hati+(1-6)hatj)#

Perform the subtraction within the vector:

#(x,y)=(1,6)+t(hati-5hatj)" [1]"#

The vector equation of a line from #(1, 6)# to #(1,3)# is:

#(x,y)=(1,6)+t((1-1)hati+(3-6)hatj)#

Perform the subtraction within the vector:

#(x,y)=(1,6)+t(-3hatj)" [2]"#

The vector equation of a line from #(1, 6)# to #(5,5)# is:

#(x,y)=(1,6)+t((5-1)hati+(5-6)hatj)#

Perform the subtraction within the vector:

#(x,y)=(1,6)+t(4hati-hatj)" [3]"#

Here are the 3 vector equations:

#(x,y)=(1,6)+t(hati-5hatj)" [1]"#
#(x,y)=(1,6)+t(-3hatj)" [2]"#
#(x,y)=(1,6)+t(4hati-hatj)" [3]"#

To obtain the new points, evaluate equations [1], [2], and [3] at t = 3

#(x,y)=(1,6)+3(hati-5hatj)#
#(x,y)=(1,6)+3(-3hatj)#
#(x,y)=(1,6)+3(4hati-hatj)#

#(x,y)=(4,-9)#
#(x,y)=(1,-3)#
#(x,y)=(13,3)#

Jim G.
May 10, 2017

#(4,-9),(1,-3),(13,3)#

Explanation:

#"let " A=(2,1),B=(1,3),C=(5,5)" and " D=(1,6)#

#"and " A',B',C'" be the images of A,B" and "C#
#"under the dilatation"#

#vec(DA)=ula-uld=((2),(1))-((1),(6))=((1),(-5))#

#rArrvec(DA')=3((1),(-5))=((3),(-15))#

#rArrA'=(1+3,6-15)=(4,-9)#
#color(blue)"--------------------------------------------------"#

#vec(DB)=ulb-uld=((1),(3))-((1),(6))=((0),(-3))#

#rArrvec(DB')=3((0),(-3))=((0),(-9))#

#rArrB'=(1+0,6-9)=(1,-3)#
#color(blue)"--------------------------------------------------"#

#vec(DC)=ulc-uld=((5),(5))-((1),(6))=((4),(-1))#

#rArrvec(DC')=3((4),(-1))=((12),(-3))#

#rArrC'=(1+12,6-3)=(13,3)#
#color(blue)"-------------------------------------------------"#

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