# Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof ### Rhombus in Rectangle

##### Age 14 to 16Challenge Level

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus. ### Fitting In

##### Age 14 to 16Challenge Level

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . . ### Towering Trapeziums

##### Age 14 to 16Challenge Level

Can you find the areas of the trapezia in this sequence? ### Round and Round

##### Age 14 to 16Challenge Level

Prove that the shaded area of the semicircle is equal to the area of the inner circle. ### Folding Squares

##### Age 14 to 16Challenge Level

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced? ### A Chordingly

##### Age 11 to 14Challenge Level

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle. ### Pythagoras Proofs

##### Age 14 to 16Challenge Level

Can you make sense of these three proofs of Pythagoras' Theorem? ### Folding Fractions

##### Age 14 to 16Challenge Level

What fractions can you divide the diagonal of a square into by simple folding? ### Zig Zag

##### Age 14 to 16Challenge Level

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line? ### Pent

##### Age 14 to 18Challenge Level

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus. ### Lens Angle

##### Age 14 to 16Challenge Level

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees. ### Pythagorean Triples I

##### Age 11 to 16

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it! ### The Pillar of Chios

##### Age 14 to 16Challenge Level

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle. ### Pythagorean Triples II

##### Age 11 to 16

This is the second article on right-angled triangles whose edge lengths are whole numbers. ### Pareq Exists

##### Age 14 to 16Challenge Level

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines. ### Circle Box

##### Age 14 to 16Challenge Level

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit? ### Kite in a Square

##### Age 14 to 16Challenge Level

Can you make sense of the three methods to work out the area of the kite in the square? ### Three Balls

##### Age 14 to 16Challenge Level

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle? ### Long Short

##### Age 14 to 16Challenge Level

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle? ### Coins on a Plate

##### Age 11 to 14Challenge Level

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle. ### Cosines Rule

##### Age 14 to 16Challenge Level

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement. ### Salinon

##### Age 14 to 16Challenge Level

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter? ### Gift of Gems

##### Age 14 to 16Challenge Level

Four jewellers share their stock. Can you work out the relative values of their gems? ### Matter of Scale

##### Age 14 to 16Challenge Level

Prove Pythagoras' Theorem using enlargements and scale factors. ### Cyclic Quad Jigsaw

##### Age 14 to 16Challenge Level

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic? ### Ratty

##### Age 11 to 14Challenge Level

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation? ### Encircling

##### Age 14 to 16Challenge Level

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape? ### Triangular Intersection

##### Age 14 to 16 ShortChallenge Level

What is the largest number of intersection points that a triangle and a quadrilateral can have? ### Converse

##### Age 14 to 16Challenge Level

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true? ### Parallel Universe

##### Age 14 to 16Challenge Level

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD. ### No Right Angle Here

##### Age 14 to 16Challenge Level

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other. ### Disappearing Square

##### Age 11 to 14Challenge Level

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . . ### Similarly So

##### Age 14 to 16Challenge Level

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD. ### Angle Trisection

##### Age 14 to 16Challenge Level

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square. ### Find the Fake

##### Age 14 to 16Challenge Level

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin? ### Same Length

##### Age 11 to 16Challenge Level

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it? ### Square Mean

##### Age 14 to 16Challenge Level

Is the mean of the squares of two numbers greater than, or less than, the square of their means? ### Chameleons

##### Age 11 to 14Challenge Level

Whenever two chameleons of different colours meet they change colour to the third colour. Describe the shortest sequence of meetings in which all the chameleons change to green if you start with 12. . . . ### Unit Fractions

##### Age 11 to 14Challenge Level

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation. ### More Number Sandwiches

##### Age 11 to 16Challenge Level

When is it impossible to make number sandwiches? ### Picturing Pythagorean Triples

##### Age 14 to 18

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself. ### Go Forth and Generalise

##### Age 11 to 14

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important. ### Logic

##### Age 7 to 14

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article. ### Cross-country Race

##### Age 14 to 16Challenge Level

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places? ### Picture Story

##### Age 14 to 16Challenge Level

Can you see how this picture illustrates the formula for the sum of the first six cube numbers? ### Mouhefanggai

##### Age 14 to 16

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai. ### Concrete Wheel

##### Age 11 to 14Challenge Level

A huge wheel is rolling past your window. What do you see? ### Tessellating Hexagons

##### Age 11 to 14Challenge Level

Which hexagons tessellate? ### Natural Sum

##### Age 14 to 16Challenge Level

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . . ### Children at Large

##### Age 11 to 14Challenge Level

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?