A line segment has endpoints at (3 ,2 ) and (5 ,4 ). The line segment is dilated by a factor of 4 around (2 ,3 ). What are the new endpoints and length of the line segment?

2 Answers
P dilip_k
Oct 13, 2016

drawn
We know,if a point P of coordinate (a,b) be dilated by a factor n around the point of coordinate (h,k), then after dilation the new position of point will be P'equiv(n(a-h)+h,n(b-k)+k)

This means

P(a,b)stackrel("dilated nX arnd"(h,k))->P^'(n(a-h)+h,n(b-k)+k)

Using this formula we get

A(3,2)stackrel("dilated 4X arnd"(2,3))->A^'(4(3-2)+2,4(2-3)+3)=A^'(6,-1)

B(5,4)stackrel("dilated 4X arnd"(2,3))->B^'(4(5-2)+2,4(4-3)+3)=B^'(14,7)

Length of

AB=sqrt((3-5)^2+(2-4)^2)=2sqrt2

Length of

A'B'=sqrt((14-6)^2+(7+1)^2)=8sqrt2

Cesareo R.
Oct 13, 2016

(6,-1) and (14,7)
length = 8sqrt(2)

Explanation:

If p_0=(2,3) is the dilation center, then p_1=(3,2) and p_2 = (5,4) after the dilation by a factor lambda their position will be

p'_1=p_0+lambda(p_1-p_0)
p'_2=p_0+lambda(p_2-p_0)

so if lambda= 4

p'_1=(2,3)+4(3-2,2-3)=(6,-1)
p'_2=(2,3)+4(5-2,4-3)=(14,7)

and

norm(p'_1-p'_2)=8sqrt(2)

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