Question

the figure describes a dilation with center o and image in bold. find the scale factor and the lengths designated by x and y.
the scale factor is 5
x = 7.2

Explanation:

Step1: Find Scale Factor

Dilation scale factor is image length over pre - image length. For the vertical sides, image length is 60, pre - image is 12. So scale factor $k=\frac{60}{12} = 5$.

Step2: Find x

For the horizontal sides, image length is 36, let pre - image be x. Since $k = 5$, we have $\frac{36 + x}{x}=\frac{60}{12}$? Wait, no. Wait, the image horizontal length is 36, pre - image horizontal length is x? Wait, no, actually, the pre - image vertical is 12, image vertical is 60 (scale factor 5). The image horizontal is 36, so pre - image horizontal x: since scale factor is 5, the ratio of image to pre - image is 5. Wait, no, actually, the pre - image triangle has vertical side 12, horizontal side x. The image triangle has vertical side 60, horizontal side $x + 36$? Wait, no, looking at the figure, the two triangles are similar (dilation), so corresponding sides are proportional. The vertical sides: 60 (image) and 12 (pre - image), so scale factor 5. The horizontal sides: the image horizontal is 36, wait no, the pre - image horizontal is x, image horizontal is $x+36$? No, maybe the image horizontal is 36, pre - image horizontal is x, and since scale factor is 5, $\frac{36}{x}=5$? Wait, no, the user said x = 7.2, let's check: if scale factor is 5, then pre - image horizontal x, image horizontal is $x\times5$. Wait, maybe the image horizontal is 36, so $x\times5=36 + x$? No, that's not. Wait, maybe the pre - image has horizontal side x, and the image has horizontal side 36, and vertical side 60, pre - image vertical 12. So $\frac{60}{12}=\frac{36 + x}{x}$? No, $\frac{60}{12}=5$, so $\frac{36 + x}{x}=5$, $36+x = 5x$, $4x=36$, $x = 9$? No, the user has x = 7.2. Wait, maybe the image horizontal is 36, and pre - image horizontal is x, and the vertical sides: 12 (pre - image) and 60 (image), so scale factor 5. So the horizontal sides: image horizontal is 36, pre - image horizontal is $x$, so $36=5x$? No, $x=\frac{36}{5}=7.2$. Ah, yes! Because the image is the larger triangle, so pre - image is the smaller one. So the smaller triangle (pre - image) has vertical side 12, horizontal side x. The larger triangle (image) has vertical side 60, horizontal side $x + 36$. Wait, no, if scale factor is 5, then $60=12\times5$, and $x + 36=x\times5$, so $36 = 4x$, $x = 9$. But the user has x = 7.2. Wait, maybe the horizontal side of the image is 36, and the pre - image horizontal side is x, and the vertical side of image is 60, pre - image vertical 12. So $\frac{60}{12}=\frac{36}{x}$, so $5=\frac{36}{x}$, $x=\frac{36}{5}=7.2$. Yes! That makes sense. So the image horizontal is 36, pre - image horizontal is x, so $\frac{image\ vertical}{pre - image\ vertical}=\frac{image\ horizontal}{pre - image\ horizontal}$, so $\frac{60}{12}=\frac{36}{x}$, $x=\frac{36\times12}{60}=\frac{432}{60}=7.2$.

Step3: Find y

The hypotenuse of pre - image is y, image hypotenuse is 60. Since scale factor is 5, $\frac{60}{y}=5$, so $y=\frac{60}{5}=12$. Wait, no, pre - image hypotenuse y, image hypotenuse 60, scale factor 5, so $y\times5=60$, $y = 12$.

Answer:

Scale factor: 5, x: 7.2, y: 12