Question

graph the image of the figure using the transformation given.

1. dilation of 2 about the origin
2. dilation of 0.5 about the origin

the polygons in each pair are similar. find the missing side length.
3)
scale factor from a to b = 3 : 4
4)
scale factor from a to b = 5 : 6
the polygons in each pair are similar. find the scale factor of the smaller figure to the larger figure.
5)
6)
the polygons in each pair are similar. find the missing side length.
7)
8)

Explanation:

Step1: Identify the scale factor relationship

For similar polygons, the ratio of corresponding sides is equal to the scale factor. Given the scale factor from A to B is \(3:4\), and the side length of A is 15. Let the missing side length of B be \(x\). So we have the proportion \(\frac{15}{x}=\frac{3}{4}\) (wait, no, scale factor from A to B is \(3:4\), so actually \(\frac{\text{Side of A}}{\text{Side of B}}=\frac{3}{4}\)? Wait, no, scale factor from A to B means if A is the original and B is the image, then \(\frac{\text{Side of B}}{\text{Side of A}}=\frac{4}{3}\) (since scale factor \(3:4\) from A to B, so B is larger by a factor of \(4/3\)). Wait, let's correct: scale factor from A to B is \(3:4\), so \(\frac{\text{Length of A}}{\text{Length of B}}=\frac{3}{4}\)? No, scale factor is usually defined as \(\frac{\text{Image}}{\text{Original}}\). So if A is original and B is image, scale factor \(k = \frac{4}{3}\) (since \(3:4\) from A to B, so B is \(4/3\) times A). Wait, the problem says "scale factor from A to B = \(3:4\)". So let's set up the proportion: \(\frac{\text{Side of A}}{\text{Side of B}}=\frac{3}{4}\)? No, that would mean B is smaller, but if scale factor from A to B is \(3:4\), then B is larger. Wait, maybe the notation is \(\frac{\text{Length of B}}{\text{Length of A}}=\frac{4}{3}\). Wait, let's take the given: side of A is 15, find side of B. So \(\frac{15}{x}=\frac{3}{4}\) is wrong. Wait, scale factor from A to B is \(3:4\), so \(\frac{\text{A's side}}{\text{B's side}}=\frac{3}{4}\), so solving for B's side: \(x=\frac{15\times4}{3}\).

Step2: Calculate the missing side

\(x = \frac{15\times4}{3}= 20\). Wait, let's check: \(15\div3 = 5\), \(5\times4 = 20\). So the missing side length is 20.

Wait, let's do problem 3:

Given similar polygons, scale factor from A to B is \(3:4\), side of A is 15, find side of B.

Since scale factor from A to B is \(3:4\), that means \(\frac{\text{Length of A}}{\text{Length of B}}=\frac{3}{4}\)? No, that would mean B is larger. Wait, scale factor is often \(\frac{\text{Image}}{\text{Original}}\). So if A is original, B is image, scale factor \(k=\frac{4}{3}\), so length of B = length of A \(\times k\). So length of B = \(15\times\frac{4}{3}=20\). Yes, that makes sense. So the missing side length is 20.

For problem 4:

Scale factor from A to B is \(5:6\), side of A is 25, find side of B.

So length of B = \(25\times\frac{6}{5}= 30\). Let's check: \(\frac{25}{30}=\frac{5}{6}\), which matches the scale factor \(5:6\) from A to B.

For problem 5:

Polygons are similar. Smaller figure has sides 8,8; larger has 12,9. Wait, first, find corresponding sides. Let's see, smaller figure: sides 8 (vertical) and 8 (horizontal). Larger figure: vertical side 9? Wait, no, maybe the sides are 8 (vertical) and 8 (horizontal) for smaller, and 12 (horizontal) and 9 (vertical) for larger. Wait, similar polygons, so the ratio of corresponding sides should be equal. So \(\frac{8}{9}=\frac{8}{12}\)? No, that's not equal. Wait, maybe the smaller figure has sides 8 (vertical) and 8 (horizontal), larger has 12 (horizontal) and let's say vertical side \(y\). Wait, no, the problem is "Find the scale factor of the smaller figure to the larger figure". So first, find the ratio of corresponding sides. Let's find the ratio of the horizontal sides: smaller horizontal is 8, larger is 12. So ratio is \(8:12 = 2:3\). Check vertical sides: smaller vertical is 8, larger is 9? Wait, that can't be. Wait, maybe the smaller figure has sides 8 (vertical) and 8 (horizontal), larger has 12 (horizontal) and 9 (vertical)? No, that would not be similar. Wait, maybe I misread. Let's look again: problem 5: smaller figure has sides 8 (left) and 8 (bottom), larger has bottom 12 and right 9. Wait, maybe the corresponding sides are 8 (bottom of small) and 12 (bottom of large), and 8 (left of small) and 9 (right of large)? No, that would be \(8/12 = 2/3\) and \(8/9\), which are not equal. Wait, maybe the smaller figure is a square (8x8) and the larger is a rectangle (12x9)? No, that's not similar. Wait, maybe I made a mistake. Wait, similar polygons have all corresponding sides in proportion. So let's check the ratios. If smaller has sides 8 and 8, larger has 12 and 9. Then \(8/12 = 2/3\), \(8/9\) is not equal. So maybe the sides are 8 (left) and 12 (left of large)? Wait, no, the figure: first polygon (smaller) has left side 8, bottom 8; second (larger) has bottom 12, right side 9. Wait, maybe the corresponding sides are left side of small (8) and right side of large (9), and bottom of small (8) and bottom of large (12). Then \(8/9\) and \(8/12 = 2/3\), not equal. So maybe the scale factor is based on the sides that are given. Wait, maybe the smaller figure has sides 8 (bottom) and 8 (left), larger has bottom 12 and left side... Wait, maybe the right side of large is 9, so left side of large should be in proportion. Wait, maybe the smaller is a rhombus (all sides 8) and the larger is a parallelogram with bottom 12 and right side 9. Then the scale factor would be \(8/12 = 2/3\) for bottom, and \(8/9\) for right side, which is not equal. So maybe I misread the problem. Wait, the problem says "The polygons in each pair are similar. Find the scale factor of the smaller figure to the larger figure." So let's take the given sides: smaller has 8 (let's say side 1) and 8 (side 2), larger has 12 (side 1) and 9 (side 2). Wait, no, that's not similar. Wait, maybe the smaller figure has sides 8 and 8, larger has 12 and 12? No, the right side is 9. Wait, maybe the figure is a quadrilateral with two sides 8 (smaller) and two sides 12 and 9 (larger). Wait, perhaps the correct corresponding sides are 8 (smaller) and 12 (larger) for one pair, and 8 (smaller) and 9 (larger) for another? No, that's not possible. So maybe the problem is that the smaller figure has sides 8 (bottom) and 8 (left), larger has bottom 12 and left side \(y\), and right side 9. Then \(8/12 = 2/3\), so \(8/y = 2/3\) => \(y = 12\), but right side is 9, which is not 12. So maybe I made a mistake in the problem number. Wait, the user asked to solve one sub-question, let's pick problem 3.

Answer:

For problem 3, the missing side length is \(\boldsymbol{20}\). (If we solve problem 3: scale factor from A to B is \(3:4\), side of A is 15. Let \(x\) be side of B. Then \(\frac{15}{x}=\frac{3}{4}\) (wait, no, scale factor from A to B is \(3:4\), so \(\frac{\text{Side of A}}{\text{Side of B}}=\frac{3}{4}\) is incorrect. Correctly, scale factor from A to B is \(\frac{\text{Side of B}}{\text{Side of A}}=\frac{4}{3}\), so \(x = 15\times\frac{4}{3}=20\).)