geometry

The endpoints of are A(9, 4) and B(5, –4). The endpoints of its image after a dilation are A'(6, 3) and B'(3, –3). can you please Explain how to find the scale factor.

1. 👍
2. 👎
3. 👁
1. I had to puzzle this out myself. In this case, we need to
(a) find the center of dilation
(b) verify that the dilation of A to A' is the same as that of B to B'

Since dilation is a linear scaling, both A and B are moving toward some point C. Naturally, if the dilation were 0, both points would contract to the same point: the intersection of the lines AA' and BB'

The line containing A and A' is (y-3)/(x-6) = 1/3
The line containing B and B' is (y+3)/(x-3) = -1/2

These lines intersect at (-3,0)

So, now, let's verify that the scale factor is the same along both directions

CA'/CA = √90/√160 = 3/4
CB'/CB = √45/√80 = 3/4

So, the scale factor is 3/4, using (-3,0) as the center of dilation.

1. 👍
2. 👎
2. Did you notice that slope AB = slope A'B' = 2 ?

So by extending AA' and BB' until they meet we can find the centre of dilation
Making a neat graph shows that this centre of dilation is C(-3,0)
or
You can find the equation of AA',which was y = (1/3)x + 1, and the equation of BB', which was y = (-1/2)x - 3/2
solving these two to get (-3,0)

AC = √(144+16) = √160 = 4√10
A'C = √(81+9) = √90 = 3√10
A'C/AC = 3V10/(4√10) = 3/4

BC = √(64+16) = √80 = 4√5
B'C = √(36+9) = √45 = 3√5
B'C/BC = 3√5/(4√5) = 3/4

scale factor is 3/4

1. 👍
2. 👎
3. Seems like we were on some kind of Vulcan mind-meld.
Even our choice of C for the centre of dilation was the same, spooky!
I had never seen that kind of question, and actually printed myself out a sheet of graph paper, lol

1. 👍
2. 👎
4. money baby

1. 👍
2. 👎
5. Dang! Steve is mega smart! He be answering all of the questions I look up.

1. 👍
2. 👎

Similar Questions

1. calculas

Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval. (Round your answers to four decimal

2. geometry

find the midpoint of the segment with the following endpoints (7,3) and (2,9)

3. geometry

If a line segment has endpoints A(3x+5, 3y) and B(x-1, -y) what are the coordinates of the midpoint AB?

4. Geometry

1.) what is the length of the line segment whose endpoints are (1,1) and (3,-3)? 2.) what are the coordinates of the midpoint of the line segment whose endpoints are (c,0) and (0,d)?

1. Geometry

What is the length of a segment with endpoints at (–2, 5) and (4, 5)?

2. geometry

Trisha drew a pair of line segments starting from a vertex. Which of these statements best compares the pair of line segments with the vertex? Answer A:Line segments have two endpoints and a vertex is a common endpoint where two

3. Geometry

Points A (-10,-6) and B (6,2) are the endpoints of AB. What are the coordinates of point C on AB such that AC is 3/4 the length of AB? a. (0,-1) b. (2,0) c. (-2,-2) d. (4,1)

4. Geometry

What is the length of the line segment whose endpoints are 1-4 and9,2

1. Calculus

a) Estimate the area under the graph of f(x)=7+4x^2 from x=-1 to x=2 using three rectangles and right endpoints. R3= ???? Then improve your estimate by using six rectangles. R6= ????? Sketch the curve and the approximating

2. geometry

Which of these is a correct step in constructing congruent line segments? a) use a straightedge to draw two equal arcs from the endpoints. b) use a compass to join the endpoints of the line segment. c) use a straightedge to

3. Geometry

A segment has endpoints X (-6,2) and Y (-1,-3). Which are the coordinates of the image for a dilation with scale factor of 1.5 and center at the origin? A. X'(-7.5) Y (-2.5,-5.5) B. X'(-4.5) Y (.5,-5.5) C. X'(-3,1) Y (-.5,-1.5) D.

4. Geometry

A square graphed on the coordinate plane has a diagonal with endpoints E (2,3) and F (0,-3) what are the coordinates of the endpoints of the other diagonal?