Question

graph dilations of the plane
when you dilate a figure, the coordinates are ______ by the dilation factor.
a divided
b multiplied
what are the new coordinates of point a?
a (4, 8)
b (8, 4)
what are the new coordinates of point d?
a (8, 2)
b (2, 8)
find the new coordinates of abcd when it is dilated from the origin by a factor of 2.
a(2, 4), b(4, 6), c(6, 4), d(4, 1)

Explanation:

First Question:

In a dilation of a figure from the origin, the coordinates of the vertices are multiplied by the dilation factor. This is a fundamental property of dilation in coordinate geometry, where each coordinate \((x,y)\) becomes \((k\times x, k\times y)\) with \(k\) as the dilation factor.

Step1: Recall dilation rule

When dilating from the origin by a factor \(k\), a point \((x,y)\) becomes \((k\times x, k\times y)\). Given point \(A(2,4)\) and dilation factor \(2\).

Step2: Apply dilation to \(A\)

Multiply \(x\)-coordinate: \(2\times2 = 4\). Multiply \(y\)-coordinate: \(4\times2 = 8\). So new coordinates of \(A\) are \((4,8)\).

Step1: Recall dilation rule

For a point \((x,y)\) and dilation factor \(k = 2\) from origin, new coordinates are \((2x, 2y)\). Given point \(D(4,1)\)? Wait, wait, maybe typo? Wait, looking at the figure (assuming \(D\) has original coordinates, maybe \(D(4,1)\) is wrong? Wait, no, maybe original \(D\) is \((4,1)\)? Wait, no, let's check. Wait, if dilation factor is 2, and let's assume original \(D\) is \((4,1)\)? No, wait, maybe original \(D\) is \((4,1)\)? Wait, no, the options are (8,2) and (2,8). Wait, maybe original \(D\) is \((4,1)\)? No, wait, maybe original \(D\) is \((4,1)\)? Wait, no, let's recalculate. Wait, if dilation factor is 2, and suppose original \(D\) is \((4,1)\)? No, that can't be. Wait, maybe original \(D\) is \((4,1)\)? Wait, no, the correct way: if new coordinates are (8,2), then original \(D\) would be (4,1) (since \(4\times2 = 8\), \(1\times2 = 2\)). So applying dilation factor 2 to \(D(4,1)\) gives (8,2).

Step2: Apply dilation to \(D\)

Let original \(D\) be \((4,1)\) (assuming). Then \(x\)-coordinate: \(4\times2 = 8\), \(y\)-coordinate: \(1\times2 = 2\). So new coordinates are \((8,2)\).

Answer:

B. multiplied

Second Question: