Question

Find the scale factor

Answer 1

`(2)/(3)`

Explanation:

Method 1

To find the scale factor of the dilation of a figure, simply divide the x-value (or y-value) of a vertex of the dilated image Q'R'S'T' by the x-value (or y-value) of the corresponding vertex of the pre-image QRST.

`\implies \sf Scale\;factor=(x_(Q'))/(x_(Q))=(-2)/(-3)=(2)/(3)`

`\implies \sf Scale\;factor=(y_(T'))/(y_(T))=(4)/(6)=(2)/(3)`

Therefore, the scale factor is 2/3.

Method 2

To find the scale factor of the dilation of a figure, first find the lengths of corresponding sides using the distance formula.

`\boxed{\begin{minipage}{7.4 cm}\underline{Distance between two points}\\\\$d=√((x_2-x_1)^2+(y_2-y_1)^2)$\\\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the two points.\\\end{minipage}}`

From inspection of the given diagram:

• Q = (-3, 9)
• R = (3, 6)

`\implies QR=√((x_R-x_Q)^2+(y_R-y_Q)^2)`

`\implies QR=√((3-(-3))^2+(6-9)^2)`

`\implies QR=√((6)^2+(-3)^2)`

`\implies QR=√(36+9)`

`\implies QR=√(45)`

`\implies QR=3√(5)`

From inspection of the given diagram:

• Q' = (-2, 6)
• R' = (2, 4)

`\implies Q'R'=\sqrt{(x_(R')-x_(Q'))^2+(y_(R')-y_(Q'))^2}`

`\implies Q'R'=√((2-(-2))^2+(4-6)^2)`

`\implies Q'R'=√((4)^2+(-2)^2)`

`\implies Q'R'=√(16+4)`

`\implies Q'R'=√(20)`

`\implies Q'R'=2√(5)`

To find the scale factor of dilation that maps QRST onto Q'R'S'T', divide the length of Q'R' by the length of QR:

`\implies (Q'R')/(QR)=(2√(5))/(3√(5))=(2)/(3)`

Therefore, the scale factor is 2/3.