A line segment goes from #(2 ,3 )# to #(4 ,1 )#. The line segment is dilated about #(0 ,1 )# by a factor of #3#. Then the line segment is reflected across the lines #x=2# and #y=-1#, in that order. How far are the new endpoints from the origin?

Answer 1

Autumn Armstrong

#color(purple)("Distances of A & B after dilation and reflection "#

#color(green)(sqrt 85, sqrt73 " respectivelyy."#

#A (2, 3), B (4,1), " dilated by factor 3 about " C(0,1)#

#A(x,y) -> A'(x,y) = 3* A(x,y) - 2*C(x,y) = (3*(2,3) - 2*(0,1)) = (6,7)#

#B(x,y) -> B'(x,y) = 3* B(x,y) - 2*C(x,y) = (3*(4,1) - 2*(0,1)) = (12,1)#

#"Reflection Rule : reflect thru " x = 2, y = -1; h=2, k= -1; (2h-x, 2k-y)#

#A''(x,y) = A'((2h - x), (2k - y)) = (4-6, -2-7) = (-2, -9)#

#B''(x,y) = B'((2h - x), (2k - y)) = (4-12, -2-1) = (-8, -3)#

#OA'' = sqrt(-2^2 + -9^2) = sqrt85#

#OB'' = sqrt(-8^2 + -3^2) = sqrt 73#

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email

Answer 2

Jayden Jackson

The new endpoints of the line segment after dilation about the point (0, 1) by a factor of 3 would be (6, 0) and (12, -2). After reflecting across the line x = 2, the new endpoints become (10, 0) and (4, -2). Finally, after reflecting across the line y = -1, the new endpoints become (10, -2) and (4, -4). To find the distance of these points from the origin, we can use the distance formula, which is √((x2 - x1)^2 + (y2 - y1)^2). Calculating for each point:

For the point (10, -2): Distance = √((10 - 0)^2 + (-2 - 0)^2) = √(100 + 4) = √104 ≈ 10.2

For the point (4, -4): Distance = √((4 - 0)^2 + (-4 - 0)^2) = √(16 + 16) = √32 ≈ 5.7

Therefore, the distance of the new endpoints from the origin are approximately 10.2 units and 5.7 units.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email

Answer from HIX Tutor

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.