Transformations, congruence, vectors and similarity Revision Notes

Recognising Similar Triangles

Two triangles are similar if they have the same shape. The key condition for similarity is that their corresponding angles are equal. If two pairs of corresponding angles are equal, then the third pair must also be equal, guaranteeing similarity. This is known as the Angle-Angle (AA) similarity criterion. Look for parallel lines, vertically opposite angles, or angles on a straight line to help identify equal angles. For example, if two triangles share an angle and have a pair of parallel lines creating corresponding angles, they will be similar.

Corresponding Sides and Scale Factor

Once you've established that two triangles are similar, their corresponding sides are in proportion. This means the ratio of any pair of corresponding sides is constant. This constant ratio is called the scale factor. To find corresponding sides, identify the sides opposite equal angles. For example, if angle A in triangle ABC equals angle D in triangle DEF, then side BC is corresponding to side EF. If side AC corresponds to side DF and angle B equals angle E, then AC/DF is part of the scale factor." Always ensure you are consistent with the order of vertices when writing similarity statements (e.g., triangle ABC ~ triangle DEF corresponds to A with D, B with E, C with F).

Calculating Unknown Side Lengths

To find an unknown side length in a pair of similar triangles, first identify the corresponding sides. Then, set up a proportion using known side lengths and the unknown side. For instance, if triangle ABC is similar to triangle DEF, and you know AB, DE, and BC, you can set up the proportion AB/DE = BC/EF to find the unknown length EF. It is often helpful to sketch the triangles separately, orienting them the same way, to clearly see corresponding sides. The scale factor can be used either to enlarge or reduce sides.

Applications of Similar Triangles

Similar triangles are widely used in various mathematical problems. They are especially useful in geometry problems involving lengths, heights, and distances where direct measurement is difficult or impossible. For instance, you can use shadows to estimate the height of tall objects or determine distances across a river. Many problems will involve one triangle embedded within another, forming a smaller triangle and a larger one that are similar by AA similarity, often due to shared angles and parallel lines.

What are Similar Triangles?

Similar triangles have the same shape but can be different sizes. This means all their corresponding angles are equal, and the ratio of their corresponding sides is constant. This constant ratio is called the scale factor. For example, if triangle ABC is similar to triangle XYZ, then ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z, and AB/XY = BC/YZ = AC/XZ. Recognising vertically opposite angles, angles on parallel lines (alternate interior or corresponding), and common angles in overlapping triangles are key to identifying equal angles and thus similar triangles.

Conditions for Similarity

To show that two triangles are similar, you need to prove one of the following conditions:

1. Angle-Angle-Angle (AAA): All three pairs of corresponding angles are equal. In practice, if two pairs of corresponding angles are equal, the third pair must also be equal because angles in a triangle sum to 180 degrees.
2. Side-Side-Side (SSS): All three pairs of corresponding sides are in the same ratio.
3. Side-Angle-Side (SAS): Two pairs of corresponding sides are in the same ratio, and the included angles (the angles between those sides) are equal.

For foundation tier, AAA is the most commonly used method, often identified by finding just two pairs of equal angles.

Using Scale Factors to Find Unknown Sides

Once you've established that two triangles are similar, you can use the scale factor to find unknown side lengths. The scale factor is the ratio of a side length in one triangle to the corresponding side length in the other triangle. You must match corresponding sides correctly! For example, if ┼ABC ~ ┼DEF and you know AB, DE, and BC, you can find EF by setting up the proportion AB/DE = BC/EF. Rearrange to solve for the unknown. Always be careful to set up the ratios consistently, e.g., (side from larger triangle) / (corresponding side from smaller triangle).

Practical Application: Identifying Corresponding Sides

It's vital to correctly identify corresponding sides. Corresponding sides are always opposite corresponding equal angles. For instance, if ∠A = ∠P in similar triangles ┼ABC and ┼ PQR, then the side opposite ∠A (which is BC) corresponds to the side opposite ∠P (which is QR). Drawing the triangles with the same orientation or explicitly listing the corresponding vertices can help avoid errors.

What is an Enlargement?

An enlargement is a transformation that changes the size of a shape but maintains its original shape. It always requires a 'centre of enlargement' and a 'scale factor'. The centre specifies the fixed point from which the enlargement happens, and the scale factor determines how much bigger or smaller the shape becomes. Understanding these two components is crucial for performing or describing any enlargement accurately.

Centre of Enlargement

The centre of enlargement is a coordinate point that does not move during the transformation. All points on the original shape are moved away from (or towards) this centre. To find the centre of enlargement graphically, draw lines connecting corresponding vertices of the original shape and its image; these lines will intersect at the centre of enlargement. For a given point (a, b) and centre (x, y), the vector from the centre to the point is (a-x, b-y).

Scale Factor

The scale factor, often denoted by 'k', determines the ratio of corresponding lengths in the image to the original shape. If k > 1, the shape gets larger. If 0 < k < 1, the shape gets smaller (a reduction). If k < 0, the shape is enlarged and inverted through the centre of enlargement. To find the new coordinates (a', b') of a point (a, b) with respect to centre (x, y) and scale factor k, use the formula: (a', b') = (x + k(a - x), y + k(b - y)). This means the vector from the centre to the new point is k times the vector from the centre to the original point.

Performing an Enlargement

To enlarge a shape with a given centre and scale factor: 1. Draw straight lines from the centre of enlargement through each vertex of the original shape. 2. Measure the distance from the centre to each vertex. 3. Multiply each distance by the scale factor. 4. Measure this new distance along the line from the centre to find the new vertex. 5. Connect the new vertices to form the enlarged image. If the scale factor is negative, measure in the opposite direction from the centre along the lines you drew.

What is Enlargement?

Enlargement is a geometric transformation that changes the size of a shape. It does not change the shape's orientation or internal angles, meaning the enlarged shape is similar to the original. Every enlargement requires two pieces of information: a scale factor and a centre of enlargement. Understanding these components is fundamental to accurately performing and describing enlargements. The term 'image' refers to the shape after enlargement, and 'object' refers to the original shape. When a shape is enlarged, all corresponding lengths change by the scale factor, but angles remain the same.

Scale Factor

The scale factor determines how much larger or smaller the image will be compared to the object. If the scale factor is greater than 1, the image will be larger than the object. If the scale factor is a fraction between 0 and 1, the image will be smaller than the object (often called a 'reduction'). A scale factor of 1 means the image is identical to the object. A negative scale factor indicates that the enlargement happens on the opposite side of the centre of enlargement, and the image will be inverted. For example, a scale factor of -2 means the image is twice as large, inverted, and on the opposite side of the centre.

Centre of Enlargement

The centre of enlargement is a fixed point from which all distances to the vertices of the object are measured and then scaled. To enlarge a shape, draw straight lines from the centre of enlargement through each vertex of the original shape. The new vertices of the enlarged shape will lie along these lines. The distance from the centre to each new vertex is found by multiplying the distance from the centre to the corresponding original vertex by the scale factor. If a point is at coordinates (x, y) and the centre of enlargement is (a, b) with a scale factor of k, the new coordinates (x', y') can be found using the formula: x' = a + k(x - a) and y' = b + k(y - b). While this formula is useful, many foundation-level enlargements can be done by counting squares or using construction lines.