Question

Rectangle ABCD has vertices A(1, 2) , B(4, 2) , C(1, −2) , and D(4, −2) . A dilation with a scale factor of 6 and centered at the origin is applied to the rectangle.

Which vertex in the dilated image has coordinates of (24, 12)

A.A′

B.B′

C.C′

D.D′

Answer 1

Answer:

B.B′

Step-by-step explanation:

In a dilation, the coordinates of each point are multiplied by the scale factor.

For a point to have an image at (24, 12) with a scale factor of 6, we would have begun at

(24/6, 12/6) = (4, 2)

These are the coordinates of B, so the image point would be B'.

Answer 2

The vertex in the dilated image whose coordinates would be (24,12) is given by: Option B: B'

How dilation of a rectangle works?

Dilation of a figure will leave its sides get scaled (multiplied) by same number.

Dilation if, is from the center of the rectangle, then distance of all the points of the rectangle from its center will get scaled by  that dilation factor.

What is the distance between two points ( p,q) and (x,y)?

The shortest distance(straight line segment's length connecting both given points) between points ( p,q) and (x,y) is:

$D = \sqrt{(x-p)^2 + (y-q)^2} \: \rm units$

For this case, we're provided that:

The vertices of the original rectangle are:

A(1, 2) , B(4, 2) , C(1, −2) , and D(4, −2)

The center of the original rectangle is centered at origin (0,0)

A dilation of this rectangle occurs with scale factor of 6, and center of dilation being the center of the rectangle.

That means, all the points of the rectangle would move 6 times away from its center.

Let the vertex's coordinate be (x,y) which reaches to (24,12).

Since the center of the rectangle is at origin and it was center of dilation, so it won't move.

Also, we will have:

Distance of (24,12) from origin (0,0) = 6 × Distance of (x,y) from origin (0,0)

Or, using the distance formula gives us:

$\sqrt{(24 -0)^2 + (12-0)^2} = 6\sqrt{(x -0)^2 + (y-0)^2}\\ 12\sqrt{5} = 6\sqrt{x^2 + y^2}\\\\\text{Squaring both the sides}\\\\36(x^2 + y^2) = 720\\\\x^ 2 + y^2 = 20$

Checking all the vertices to see which one satisfies this relation:

• Case 1: A(1,2):

$x^2 + y^2 = 1^2 + 2^2 = 5 \neq 20$

• Case 2: B(4,2):

$x^2 + y^2 = 4^2 + 2^2 = 20$

• Case 3: C(1,-2):

$x^2 + y^2 = 1^2 + (-2)^2 = 5 \neq 20$

• Case 4: D(4,-2):

$x^2 + y^2 = 4^2 + (-2)^2 = 20$

So its either vertex B or D.

Now, as the point (24,12) is in first quadrant, and as the point B is also in first quadrant, and as dilation would make points go far from center but not changing quadrants, and that D is not in first quadrants, thus, it was vertex B who reached at (24,12) after dilation.

Thus, the vertex in the dilated image whose coordinates would be (24,12) is given by: Option B: B'

Learn more about dilation here:

https://brainly.com/question/3266920