Congruence, Circle Theorems and Transformations

Congruence, the circle theorems and transformations are all about what stays the same when a shape moves.

When are two shapes congruent, and how do you prove it?

Two triangles are congruent when they are identical: same shape and same size, however they are turned. Three measurements are enough to prove it, if they are the right three. There are four tests. SSS: three matching sides. SAS: two sides and the angle between them. ASA: two angles and the side between them. RHS: a right angle, the hypotenuse and one other side.

Similar means the same shape at any size. Two equal angles are enough to prove two triangles similar, because the third angle is then equal as well. Every pair of matching sides is then in the same ratio, the scale factor.

Area is a product of two lengths, so it scales by the square of the scale factor. Volume is a product of three lengths, so it scales by the cube. Lengths in the ratio 2 : 3 give areas in the ratio 4 : 9 and volumes in the ratio 8 : 27. See Congruence tests, Similarity conditions, Ratio of areas and Ratio of volumes.

What are the parts of a circle called?

A chord is a straight line joining two points on the circle. An arc is the curved part of the circle between those two points. A sector is the slice between two radii. A segment is the piece cut off by a chord, on the side away from the center.

A sector is a fraction of the whole circle, and the fraction is its angle divided by 360. A sector with an angle of 60° is $\frac{60}{360}=\frac{1}{6}$60/360 = 1/6 of the circle, so its area is $\frac{1}{6}$1/6 of $\pi r^2$πr² and its arc is $\frac{1}{6}$1/6 of $2\pi r$2πr.

To find the area of a segment, find the area of the sector, then subtract the triangle made by the two radii and the chord. For a quarter circle of radius 10, the sector is $\frac{1}{4}\times \pi \times 10^2=78.5$1/4 × π × 10² = 78.5 and the triangle is $\frac{1}{2}\times 10\times 10=50$1/2 × 10 × 10 = 50, so the segment is 28.5. See Chords, arcs, sectors and segments, Arcs and sectors and The area of a segment.

The circle theorems, and why they are true

The angle at the center is twice the angle at the circumference when both stand on the same arc. Most of the other theorems follow from this one.

Three theorems follow directly. The angle in a semicircle is 90°, because the angle at the center is 180°. Angles in the same segment are equal, because each is half of the same angle at the center. Opposite angles of a cyclic quadrilateral add to 180°, because their two angles at the center make the full 360°.

A tangent touches the circle at one point and meets the radius there at a right angle. Two tangents from the same outside point are equal in length, by RHS congruence.

The alternate segment theorem says the angle between a tangent and a chord equals the angle in the segment on the other side of the chord. Check it with a diameter as the chord: both angles are 90°.

The perpendicular from the center to a chord bisects the chord, again by RHS. This turns a chord problem into a right-angled triangle, and Pythagoras finishes it. See Angle at the center, Angle in a semicircle, Angles in the same segment, Cyclic quadrilaterals, Tangent properties, The alternate segment theorem and Chords and the center.

What are the four transformations?

A reflection flips a shape across a mirror line. Reflecting in the y-axis changes the sign of x: (4, 3) lands on (−4, 3).

A rotation turns a shape about a fixed point, the center. A full description needs the center, the angle and the direction. A quarter turn counterclockwise about the origin sends (x, y) to (−y, x).

A translation slides a shape without turning it, by a column vector [a, b]: a across, then b up. From (1, 2), the vector [3, −1] means 3 right and 1 down, so the point lands on (4, 1).

An enlargement changes the size. Draw a ray from the center through each vertex; the image vertex is k times as far along the ray, where k is the scale factor. A scale factor between 0 and 1 makes the shape smaller and is still called an enlargement.

The order of rotational symmetry counts how many times a shape lands on itself in one full turn. A rectangle has order 2 and an equilateral triangle has order 3.

Two transformations in a row can often be replaced by one. Two reflections in intersecting lines make a rotation, and two reflections in parallel lines make a translation of twice the gap between the lines. See Reflection, Rotation, Translation of shapes, Enlargement with a center, Rotational symmetry and Combined transformations.

How does a matrix perform a transformation?

A 2 × 2 matrix moves every point of the plane and keeps the origin fixed. Its first column is where the point (1, 0) lands, and its second column is where (0, 1) lands, so you can write the matrix down from a description. A quarter turn counterclockwise sends (1, 0) to (0, 1) and (0, 1) to (−1, 0), so the matrix has columns (0, 1) and (−1, 0).

The determinant ad − bc tells you what happens to area. Its size is the factor by which every area is multiplied, and its sign says whether the plane was flipped over. A reflection has determinant −1: area unchanged, orientation reversed. A determinant of 0 means the plane has been squashed onto a line.

Doing one transformation and then another is the same as multiplying the two matrices, with the second transformation written on the left. Swap the order and the product is usually a different matrix, which is why matrix multiplication is not commutative.

An invariant point is one the transformation sends back to where it was: a reflection fixes every point on its mirror line, and a rotation fixes only its center.

A 3 × 3 matrix works the same way in three dimensions, and its determinant is the factor by which volume is multiplied. Three linear equations in x, y and z are three planes: if the determinant is not 0 they meet at one point, and if it is 0 they share a line or have no common point. See Matrices as transformations of the plane, The determinant as an area scale factor, Composing transformations by multiplying matrices, Invariant points and lines, Matrices as transformations of space, The determinant as a volume scale factor and The geometry of three planes.

The mistakes worth naming

• Using the length ratio for an area. Area scales by the square, volume by the cube.
• Describing a rotation without its center. Angle and direction alone do not fix it.
• Using SSA to prove congruence. It can make two different triangles.
• Multiplying transformation matrices in reading order. The first transformation goes on the right.

Learn this properly in the app

Every lesson linked above is a screen in Math Challenge with a diagram, a worked example and practice questions. It builds on lines, angles and Pythagoras and polygons, solids and surface area; the algebra of matrices is in matrices and Markov chains.