# Reflecting Shapes in a Mirror Line

This is a typical question from a Dulwich College 11+ Maths paper, and it asks you to draw a reflection of the triangle in the mirror line shown on the chart.

Dulwich papers tend to be a bit tricky, and this is not the easiest version of this kind of reflective symmetry question.

For a start, the mirror line is drawn at 45 degrees rather than being horizontal or vertical, and it doesn’t help that the diagram is a bit ‘squashed’, which means the mirror line is actually at around 40 degrees rather than 45!

So how should you do it?

The first thing to do is to imagine that you were looking at yourself in the mirror from, say, 30cm away.

Your reflection will appear ‘in’ the mirror, but it won’t be on the surface of the mirror, will it?

It’ll actually seem to be 30cm ‘behind’ the mirror – which is exactly the same distance as you are in front of it.

That’s important, and you’ll have to use that fact when you do the question.

The basic steps are these:

1. Plot the ‘vertices’ (or corners) of the reflected shape one by one by drawing a small cross in pencil.
2. Join them up using a ruler and pencil.

In order to plot each corner, you need to imagine that the corner is your face and that the mirror line is the mirror.

To see your reflection, you have to be standing right in front of the mirror – looking at an angle of 90 degrees to the mirror – so to ‘see’ the reflection of a corner, you have to do the same, looking at an angle of 90 degrees to the mirror line.

The distance from your face to the mirror is the same as the distance to the spot ‘behind’ the mirror where you see your reflection.

In the same way, the distance from the corner to the mirror line is the same as the distance to the spot ‘behind’ the mirror line where the reflected point should go.

If you use the diagram at the top of this article to help you, you should be able to see that the top of the triangle is one-and-a-half diagonal squares away from the mirror line.

That means you need to go another one-and-a-half diagonal squares the other side of the mirror line (continuing in the same direction) in order to plot the reflected point.

Now repeat this for the other corners of the triangle, which are four-and-a-half and three diagonal squares away from the mirror line.

Once you’ve done that, you can join up all three points using a ruler and pencil to make the reflected triangle.

Once you get the hang of it, you may not even need to plot all the corners: if it’s a simple shape like a square or a rectangle, then you might be able to draw it from scratch.

Just make sure you label the shape if the question asks you to.

And that’s it…!

# Useful Terms in Maths

Maths is complicated, but a good first step on the road to understanding it is to get to know the most useful terms. There are lists in the front of the Bond books, but here’s my own contribution. I hope it helps!

Algebra: expressions using letters to represent unknown values, eg 2(x + 3) = 16.

Angles: there are three types of angle, depending on the number of degrees.

• acute angles are between 0 and 90 degrees.
• obtuse angles are between 90 and 180 degrees.
• reflex angles are between 180 and 360 degrees.

Arc: part of the circumference of a circle.

Averages: there are three types of average, and they are all useful in different ways.

• The mean is found by adding up all the values and dividing the total by how many there are, eg the mean of the numbers 1-10 is 5.5, as 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55, and 55 ÷ 10 = 5.5.
• The mode is the most common value (or values), eg the mode of 1, 2, 2, 3, 4, 5 is 2.
• The median of an odd number of values sorted by size is the one in the middle, eg the median of the numbers 1-5 is 3. The median of an even number of values is the mean of the two numbers in the middle, eg the median of the numbers 1-10 is 5.5, as 5 and 6 are the numbers in the middle, and 11 ÷ 2 = 5.5.

Chord: a straight line drawn between two points on the circumference of a circle.

Circumference: the distance all the way round the edge of a circle.

Congruent: triangles are congruent if they are the same shape and size, eg two right-angled triangles with sides of 3cm, 4cm and 5cm would be ‘congruent’, even if one is the mirror image of the other. You can prove that two triangles are congruent by using any of the following methods: SAS (Side-Angle-Side), SSS (Side-Side-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side) and RHS or HL (Right-angle-Hypotenuse-Side or Hypotenuse-Leg). If all three measurements of the angles and/or sides are equal, the triangles are congruent. You can only create a congruent copy of a triangle by translation, reflection or rotation. (Note: congruence is the same as similarity, except that the triangles have to be the same size.)

Cube: the result of multiplying any number by itself twice, eg 8 is the cube of 2, as 2 x 2 x 2 = 8.

Cube root: the number that has to be multiplied by itself twice to make another number, eg 2 is the cube root of 8, as 2 x 2 x 2 = 8.

Cuboid: a solid with a rectangle for each of the six sides, eg a shoe box.

Denominator: the number on the bottom of a fraction, eg 2 is the denominator of ½.

Diameter: the length of a line drawn across a circle passing through the centre.

Equation: any line of numbers and operators with an equals sign in the middle, showing that the two sides balance, eg 4x + 12 = 34.

Factor: a number that goes into another number evenly, eg 6 is a factor or 12.

Fibonacci series: a sequence of numbers created by adding the previous two numbers together to get the next one, eg 1, 1, 2, 3, 5, 8, 13…

Formula: a way of calculating the answer to a common problem using letters or words, eg the formula for distance is speed x time (or D = S x T).

Highest Common Factor (HCF): the highest number that goes into two other numbers evenly, eg the HCF of 12 and 18 is 6.

Improper fraction: a fraction that is greater than one (in other words, the numerator is greater than the denominator), eg 9/5.

Lowest Common Multiple (LCM) or Lowest/Least Common Denominator (or LCD): the lowest number that is divisible by two other numbers, eg the LCM of 6 and 8 is 24.

Multiple: a number that can be divided evenly by another number, eg 12 is a multiple of 6.

Numerator: the number on the top of a fraction, eg 3 is the numerator of ¾.

Order of operations: the sequence of doing basic mathematical sums when you have a mixture of, say, addition and multiplication. BIDMAS (or BODMAS) is a good way of remembering it, as it stands for:

• Brackets
• Indices/Order (in other words, squares, cubes and so on)
• Division
• Multiplication
• Subtraction

Note that addition doesn’t come ‘before’ subtraction – these operations have to be done in the order in which they occur in the sum, and it makes a difference to the answer, eg 4 – 3 + 2 = 3 if you do the operations in order, which is correct, but you’d get the wrong answer of -1 if you did 3 + 2 first.

Operator: the sign telling you which mathematical operation to do. The most common ones are +, -, x and ÷.

Parallel: two lines are parallel if they will never meet, eg the rails on a railway line.

Perimeter: the distance all the way round the outside of a shape.

Perpendicular: at 90 degrees to each other.

Pi (or π): a constant used to work out the circumference and area of circles, often shown as 22/7 or 3.14 although it’s actually an ‘irrational’ number, which means it goes on for ever.

Prime factors: the lowest prime numbers that can be multiplied together to make a given number, eg the prime factors of 12 are 2² x 3.

Prime numbers: a number that can only be divided by itself and one, eg 2, 3, 5, 7, 11, 13…

Probability: the chance of something happening, calculated as the number of ways of getting what you want divided by the total number of possible outcomes, eg the chance of a coin toss being heads is ½ as there is one ‘heads’ side but two sides in total. To work out the probability of a sequence of events, you have to multiply the individual probabilities together, eg the chance of a coin toss being heads twice in a row is ½ x ½ = ¼

Product: the result of multiplying two numbers together, eg 35 is the product of 5 and 7.

Quadrilateral: a four-sided shape such as the following:

• Kite: a quadrilateral with two pairs of equal sides next to each other (or ‘adjacent’ to each other).
• Parallelogram: a quadrilateral with opposite sides parallel to each other.
• Rectangle: a quadrilateral with two opposite pairs of equal sides and four right angles.
• Rhombus: a quadrilateral with equal sides.
• Square: a quadrilateral with equal sides and four right angles.
• Trapezium (or Trapezoid): a quadrilateral with one pair of parallel sides. (Note: an isosceles trapezium is symmetrical.)

Radius: the distance from the centre of a circle to the circumference.

Range: the highest minus the lowest value in a list, eg the range of the numbers 1-10 is 9.

Regular: a shape is regular if all its sides and angles are equal, eg a 50p piece is a regular (-ish!) heptagon.

Right angle: an angle of 90 degrees.

Sector: a ‘slice’ of a circle in between two radii.

Segment: a part of a circle separated from the rest by a chord.

Shapes: the name of each shape depends on the number of sides. Here are the most common ones.

• Triangles have three sides.
• Pentagons have five sides.
• Hexagons have six sides.
• Heptagons have seven sides.
• Octagons have eight sides.
• Nonagons have nine sides.
• Decagons have 10 sides.
• Hendecagons have 11 sides.
• Dodecagons have 12 sides.

Similar: triangles are similar if they are the same shape, but not necessarily the same size, eg a right-angled triangle with sides of 3cm, 4cm and 5cm is ‘similar’ to a right-angled triangle with sides of 6cm, 8cm and 10cm. (Note: similarity is the same as congruence, except that the triangles don’t have to be the same size.)

Square number: the result of multiplying any number by itself, eg 49 is a square number, as 7 x 7 = 49.

Square root: the number that has to be multiplied by itself to make another number, eg 6 is the square root of 36, as 6 x 6 = 36.

Sum: the result of adding two numbers together, eg 17 is the sum of 8 and 9.

Tangent: either a straight line that touches the circumference of a circle OR the length of the opposite side of a triangle divided by the length of the adjacent side

Transformations: there are three main kinds of transformation: reflection, rotation and translation.

• With reflection, you need to state the formula of the mirror line, eg the shape has been reflected in the line y = 4.
• With rotation, you need to state the number of degrees, the direction and the centre of rotation, eg the shape has been rotated 90 degrees clockwise around the point (4, 3).
• With translation, you need to state the change in the x and y values, eg the shape has been translated four units up and three units to the right.

Triangles: there are four main types, each with different properties.

• equilateral triangles have all three sides the same length and all three angles the same.
• isosceles triangles have two sides the same length and two angles the same.
• scalene triangles have three sides of different lengths with three different angles.
• right-angled triangles have one 90-degree angle.

Variable: an unknown in algebra, eg x or y.