Question

1. dilations use a scale factor to _ or _ the size of the pre - image. the image and the pre - image are _ figures but not _ figures.

similar figures have corresponding angles that are _ and corresponding sides that are _.
dilations with a scale factor > 1 make _ images, and dilations with a scale factor < 1 make _ images.
here are some polygons:

1. which of polygons b, c, d and e are similar to polygon a?
2. what scale factor was used to dilate polygon a to c?
3. which set of polygons have been dilated with a scale factor of 1?

triangles abc and def are similar. use this image for #5 - 8

1. find the length of segment ef.
2. find the length of segment ab.
3. what is the scale factor that takes triangle abc to def?
4. if angle b is 105 degrees then what is the size of angle e?

Explanation:

Step1: Identify similar polygons

Similar polygons have corresponding angles equal and sides proportional. By visual inspection and using the properties of dilation (which preserves shape, i.e., similarity), we check each polygon (B, C, D, E) against A.

Step2: Calculate scale factor for dilation from A to C

Count the number of units (assuming each grid - square has a side - length of 1 unit). Let's say a side of polygon A has length \(x\) and the corresponding side of polygon C has length \(y\). The scale factor \(k=\frac{y}{x}\).

Step3: Determine polygons with scale factor 1

A scale factor of 1 means the pre - image and image are congruent (same shape and size).

Step4: Use similarity of triangles for segment lengths

For similar triangles \(\triangle ABC\) and \(\triangle DEF\), if \(\frac{EF}{BC}=\frac{DF}{AC}=\frac{DE}{AB}\). Given \(BC = 9\), \(DF = 4\), \(AC=12\), \(DE = 2.5\).

• For \(EF\):

Since \(\frac{EF}{BC}=\frac{DF}{AC}\), let \(EF\) be \(x\). We have \(\frac{x}{9}=\frac{4}{12}\). Cross - multiply: \(12x=9\times4\), \(12x = 36\), \(x = 3\).

• For \(AB\):

Let \(AB\) be \(y\). Using \(\frac{DE}{AB}=\frac{DF}{AC}\), \(\frac{2.5}{y}=\frac{4}{12}\). Cross - multiply: \(4y=2.5\times12\), \(4y = 30\), \(y = 7.5\).

• For scale factor from \(\triangle ABC\) to \(\triangle DEF\):

Scale factor \(k=\frac{DF}{AC}=\frac{4}{12}=\frac{1}{3}\).

• For angle \(E\):

Similar triangles have corresponding angles equal. Since \(\angle B\) and \(\angle E\) are corresponding angles in similar triangles \(\triangle ABC\) and \(\triangle DEF\), \(\angle E=\angle B = 105^{\circ}\).

Answer:

1. Polygons C and D are similar to Polygon A.
2. The scale factor used to dilate Polygon A to C is \(2\) (assuming appropriate side - length counting from the grid).
3. No polygons (since a scale factor of 1 would mean congruent, and from the given grid - based polygons, there is no such pair shown as congruent in the context of dilation from A).
4. The length of segment \(EF\) is \(3\).
5. The length of segment \(AB\) is \(7.5\).
6. The scale factor that takes triangle \(ABC\) to \(DEF\) is \(\frac{1}{3}\).
7. The size of Angle \(E\) is \(105\) degrees.