Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof

Rhombus in Rectangle

Age 14 to 16 Challenge 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.

Towering Trapeziums

Age 14 to 16 Challenge Level:

Can you find the areas of the trapezia in this sequence?

Cosines Rule

Age 14 to 16 Challenge 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.

Lens Angle

Age 14 to 16 Challenge 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.

Zig Zag

Age 14 to 16 Challenge 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?

Age 14 to 16 Challenge 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?

Round and Round

Age 14 to 16 Challenge Level:

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

The Pillar of Chios

Age 14 to 16 Challenge 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.

Pareq Exists

Age 14 to 16 Challenge Level:

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

Triangular Intersection

Age 14 to 16 Short Challenge Level:

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

Pythagoras Proofs

Age 14 to 16 Challenge Level:

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

Find the Fake

Age 14 to 16 Challenge 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?

Fitting In

Age 14 to 16 Challenge 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. . . .

Circle Box

Age 14 to 16 Challenge 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?

Long Short

Age 14 to 16 Challenge Level:

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

Square Mean

Age 14 to 16 Challenge Level:

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

Age 14 to 16 Challenge Level:

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

Parallel Universe

Age 14 to 16 Challenge Level:

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

Pent

Age 14 to 18 Challenge 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.

Age 16 to 18 Challenge Level:

As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.

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!

Kite in a Square

Age 14 to 16 Challenge Level:

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

Pythagorean Triples II

Age 11 to 16

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

Converse

Age 14 to 16 Challenge 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?

Salinon

Age 14 to 16 Challenge 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?

Similarly So

Age 14 to 16 Challenge 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.

Shape and Territory

Age 16 to 18 Challenge Level:

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Triangle Incircle Iteration

Age 14 to 16 Challenge Level:

Keep constructing triangles in the incircle of the previous triangle. What happens?

More Number Sandwiches

Age 11 to 16 Challenge Level:

When is it impossible to make number sandwiches?

A Knight's Journey

Age 14 to 18

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

Calculating with Cosines

Age 14 to 18 Challenge Level:

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

Folding Fractions

Age 14 to 16 Challenge Level:

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

Encircling

Age 14 to 16 Challenge Level:

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

Same Length

Age 11 to 16 Challenge Level:

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

No Right Angle Here

Age 14 to 16 Challenge Level:

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

Angle Trisection

Age 14 to 16 Challenge Level:

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

Folding Squares

Age 14 to 16 Challenge 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?

Matter of Scale

Age 14 to 16 Challenge Level:

Prove Pythagoras' Theorem using enlargements and scale factors.

Three Balls

Age 14 to 16 Challenge 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?

Areas and Ratios

Age 16 to 18 Challenge Level:

Do you have enough information to work out the area of the shaded quadrilateral?

Little and Large

Age 16 to 18 Challenge Level:

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

Square Pair Circles

Age 16 to 18 Challenge Level:

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

Always Perfect

Age 14 to 16 Challenge Level:

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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.

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.

Natural Sum

Age 14 to 16 Challenge 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. . . .

The Frieze Tree

Age 11 to 16

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Picture Story

Age 14 to 16 Challenge Level:

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

Napoleon's Hat

Age 16 to 18 Challenge Level:

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?