Question

1. dilate △qrs on the coordinate plane using the origin (0, 0) as the center of dilation and a scale factor of 3 to form △qrs. label the coordinates of points q, r, and s.
2. triangle abc is graphed on a coordinate plane with vertices at a(-7, 5), b(4, 6), and c(5, 8). triangle abc is dilated by a scale factor of w with the origin as the center of dilation to create △abc. what are the coordinates of the vertices of △abc?
3. quadrilateral qrst is graphed on a coordinate plane with vertices as q(2, 14), r(10, 18), s(12, -5), and t(4, -1). quadrilateral qrst is dilated by a scale factor of 1/u with the origin as the center of dilation to create quadrilateral qrst. what are the coordinates of the vertices of quadrilateral qrst?

Explanation:

Step1: Recall dilation formula

When dilating a point $(x,y)$ with the origin $(0,0)$ as the center of dilation and a scale - factor $k$, the new coordinates $(x',y')$ are given by $(x',y')=(k\cdot x,k\cdot y)$.

Step2: Find coordinates for problem 2

From the graph, we can see that $Q(2,5)$, $R(3,3)$, $S(5,6)$. Using the dilation formula with $k = 3$, we have:
For point $Q$: $Q'(3\times2,3\times5)=(6,15)$
For point $R$: $R'(3\times3,3\times3)=(9,9)$
For point $S$: $S'(3\times5,3\times6)=(15,18)$

Step3: Find coordinates for problem 3

Given $A(-7,5)$, $B(4,6)$, $C(5,8)$ and scale - factor $w$.
For point $A$: $A'=(w\times(-7),w\times5)=(-7w,5w)$
For point $B$: $B'=(w\times4,w\times6)=(4w,6w)$
For point $C$: $C'=(w\times5,w\times8)=(5w,8w)$

Step4: Find coordinates for problem 4

Given $Q(2,14)$, $R(10,18)$, $S(12, - 5)$, $T(4,-1)$ and scale - factor $k=\frac{1}{u}$.
For point $Q$: $Q'=(\frac{1}{u}\times2,\frac{1}{u}\times14)=(\frac{2}{u},\frac{14}{u})$
For point $R$: $R'=(\frac{1}{u}\times10,\frac{1}{u}\times18)=(\frac{10}{u},\frac{18}{u})$
For point $S$: $S'=(\frac{1}{u}\times12,\frac{1}{u}\times(-5))=(\frac{12}{u},-\frac{5}{u})$
For point $T$: $T'=(\frac{1}{u}\times4,\frac{1}{u}\times(-1))=(\frac{4}{u},-\frac{1}{u})$

Answer:

Problem 2: $Q'(6,15)$, $R'(9,9)$, $S'(15,18)$
Problem 3: $A'(-7w,5w)$, $B'(4w,6w)$, $C'(5w,8w)$
Problem 4: $Q'(\frac{2}{u},\frac{14}{u})$, $R'(\frac{10}{u},\frac{18}{u})$, $S'(\frac{12}{u},-\frac{5}{u})$, $T'(\frac{4}{u},-\frac{1}{u})$