Transformations, congruence, vectors and similarity Revision Notes

Defining Enlargement

Enlargement is a geometric transformation that changes the size of a shape while preserving its overall form. It is entirely defined by two parameters: a centre of enlargement and a scale factor. The centre of enlargement is a fixed reference point from which all distances are measured. The scale factor dictates the ratio of corresponding lengths between the image and the object. A scale factor greater than 1 makes the image larger, while a scale factor between 0 and 1 makes it smaller. A negative scale factor indicates that the image is formed on the opposite side of the centre of enlargement.

Applying a Positive Scale Factor

When applying a positive scale factor, say 'k', from a centre of enlargement (C):

1. Draw straight lines from C, passing through each vertex of the object.
2. For each vertex P, measure the distance from C to P. Let this be CP.
3. The corresponding vertex P' in the image will be at a distance k * CP from C, along the same line drawn in step 1.
4. If k > 1, the image is larger than the object and further from C. If 0 < k < 1, the image is smaller and closer to C.
5. Connect the new vertices (P') to form the enlarged shape.

Applying a Negative Scale Factor

A negative scale factor, -k (where k > 0), means the image is on the opposite side of the centre of enlargement (C) compared to the object. The process is similar:

1. Draw straight lines from C, passing through each vertex of the object. Extend these lines 'backwards' through the centre of enlargement.
2. For each vertex P, measure the distance from C to P. Let this be CP.
3. The corresponding vertex P' in the image will be at a distance k * CP from C, but on the opposite side of C along the extended line.
4. If |-k| > 1, the image is larger. If 0 < |-k| < 1, the image is smaller. The image is inverted relative to the object with respect to the centre of enlargement. Connect the new vertices (P') to form the enlarged shape.

Finding the Centre of Enlargement and Scale Factor

To find the centre of enlargement given an object and its image:

1. Draw straight lines connecting corresponding vertices of the object and its image. For example, connect A to A', B to B', etc.
2. These lines will intersect at a single point. This point is the centre of enlargement.

To find the scale factor:

1. Measure the length of a side on the image and the corresponding side on the object. The scale factor is the ratio: (Image Length) / (Object Length).
2. Determine if the scale factor is positive or negative. If the image is on the same side of the centre as the object (and not inverted), the scale factor is positive. If it's on the opposite side (and inverted), the scale factor is negative.

What are Similar Shapes?

Similar shapes are figures that have exactly the same shape but possibly different sizes. Imagine taking a photo and enlarging or shrinking it without distorting anything – the original and the new photo are similar shapes. For two shapes to be similar, two conditions must be met: all corresponding angles must be equal, and all corresponding sides must be in the same proportion. This means that if you match up the angles of two similar shapes, they will be identical. If you compare the lengths of their sides, each pair of corresponding sides will have the same ratio. For example, if triangle ABC is similar to triangle PQR, then angle A = angle P, angle B = angle Q, angle C = angle R. Also, the ratio AB/PQ will be equal to BC/QR, which will be equal to AC/PR.

The Scale Factor for Lengths

The 'scale factor' is the constant ratio by which all corresponding lengths in similar shapes are related. If you have two similar shapes, one larger than the other, you can find the scale factor by dividing a length on the larger shape by the corresponding length on the smaller shape. For example, if side X in the larger shape is 15 cm and corresponding side Y in the smaller shape is 5 cm, the scale factor from smaller to larger is 15/5 = 3. This means that every side in the larger shape is 3 times longer than the corresponding side in the smaller shape. To find an unknown side length, you either multiply by the scale factor (to go from smaller to larger) or divide by the scale factor (to go from larger to smaller). Make sure you always use corresponding sides to calculate or apply the scale factor.

Identifying Corresponding Sides and Angles

When working with similar shapes, it's crucial to correctly identify corresponding sides and angles. Corresponding angles are angles that are in the same relative position in both shapes. Since corresponding angles are equal, this can often help you identify which sides correspond. For instance, the side opposite the largest angle in one shape will correspond to the side opposite the largest angle in the similar shape. Similarly, the side between two specific angles in one shape will correspond to the side between the same two angles in the other shape. Often, diagrams might be rotated, so visual inspection and angle matching are key to setting up the correct ratios for the scale factor.

What are Similar Shapes?

Similar shapes are geometric figures that have identical shapes but can differ in size. Think of a small photo and a larger print of the same photo; they're similar. For any two polygons to be similar, two conditions must be satisfied: all corresponding angles must be equal, and the ratio of all corresponding side lengths must be the same. This constant ratio is called the scale factor. If you can show that these two conditions hold, the shapes are similar. It's crucial to correctly identify corresponding angles and sides, which are the parts of the shapes that match up when one is scaled to become the other.

Using Scale Factor for Lengths

The scale factor is the key to working with similar shapes. It's the ratio by which all corresponding lengths in one shape are multiplied to get the lengths in the other shape. To find the scale factor (SF), you can divide a side length from the 'new' or image shape by the corresponding side length from the 'original' or object shape. So, SF = (length in new shape) / (corresponding length in original shape). Once you have the scale factor, you can find any unknown length in one shape by multiplying or dividing the corresponding known length in the other shape by the scale factor. If you are going from a smaller shape to a larger shape, you multiply by the scale factor. If you are going from a larger shape to a smaller shape, you divide by the scale factor (or multiply by its reciprocal).

Identifying Similar Triangles

Triangles are a common type of shape to encounter in similarity problems. There are a few ways to prove that two triangles are similar, which can simplify the process. The most common method involves showing that two pairs of corresponding angles are equal (AA similarity criterion). If two pairs of angles are equal, the third pair must also be equal because the sum of angles in a triangle is 180 degrees. Therefore, all corresponding angles are equal, satisfying the first condition for similarity. Once you know triangles are similar, you can confidently use the scale factor to find unknown side lengths.

Finding Unknown Sides in Similar Shapes

When faced with similar shapes and an unknown side length, the first step is to identify two corresponding sides for which both lengths are known. Use these to calculate the scale factor. Decide whether you are enlarging or reducing; if you are going from a smaller shape to a larger shape, the scale factor will be greater than 1. If you are going from a larger shape to a smaller shape, the scale factor will be less than 1. Then, take the known side length that corresponds to your unknown side, and either multiply by the scale factor (for enlargement) or divide by the scale factor (for reduction) to find the unknown length. Always ensure your calculation aligns with whether the unknown side is expected to be larger or smaller than its known counterpart.

Identifying Similar Shapes and Linear Scale Factor

Two shapes are similar if one can be obtained from the other by an enlargement or reduction. This means all corresponding angles are equal, and all corresponding pairs of sides are in the same ratio. This ratio is known as the linear scale factor, 'k'. To find 'k', divide a length in the larger shape by the corresponding length in the smaller shape. Ensure you consistently use the same order (e.g., larger/smaller or smaller/larger) when calculating the ratio. If shapes appear nested or overlapping, it is important to clearly distinguish between the corresponding parts.

Area Scale Factor for Similar Shapes

If the linear scale factor between two similar shapes is 'k', then the ratio of their corresponding areas is 'k'. For example, if the linear scale factor 'k' = 3, then the area of the larger shape will be 3 = 9 times the area of the smaller shape. To find an unknown area, calculate the linear scale factor first (k), then square it (k) and multiply or divide the known area by this factor. If you are given area values and need the linear scale factor, take the square root of the area ratio.

Volume Scale Factor for Similar Shapes

For 3D similar shapes (like cubes, spheres, or any other solids), if the linear scale factor between them is 'k', then the ratio of their corresponding volumes is 'k'. For instance, if the linear scale factor 'k' = 2, then the volume of the larger shape will be 2 = 8 times the volume of the smaller shape. When solving problems involving volume, carefully find the linear scale factor 'k', then cube it (k) and apply it. If you are given volume values and need the linear scale factor, take the cube root of the volume ratio.

Applying Scale Factors to Solve Problems

When solving problems with similar shapes, first identify if the problem involves lengths, areas, or volumes. Next, determine the linear scale factor 'k' using any known corresponding lengths. Then, adjust the scale factor for area problems (k) or volume problems (k). Finally, use the appropriate scale factor to calculate the unknown dimension, area, or volume. Remember to clearly state your scale factors and the dimension they refer to (linear, area, or volume) to avoid confusion. Also, be careful with units and ensure consistency.