Resources tagged with: Pythagoras' theorem

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Broad Topics > Pythagoras and Trigonometry > Pythagoras' theorem

Matter of Scale

Age 14 to 16
Challenge Level

Prove Pythagoras' Theorem using enlargements and scale factors.

Pythagoras Proofs

Age 14 to 16
Challenge Level

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

Weighty Problem

Age 11 to 14
Challenge Level

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .

Tilted Squares

Age 11 to 14
Challenge Level

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

All Tied Up

Age 14 to 16
Challenge Level

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

Tilting Triangles

Age 14 to 16
Challenge Level

A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

Equilateral Areas

Age 14 to 16
Challenge Level

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Star Gazing

Age 14 to 16
Challenge Level

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

Semi-square

Age 14 to 16
Challenge Level

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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!

Where Is the Dot?

Age 14 to 16
Challenge Level

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

In a Spin

Age 14 to 16
Challenge Level

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

Nicely Similar

Age 14 to 16
Challenge Level

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Under the Ribbon

Age 14 to 16
Challenge Level

A ribbon is nailed down with a small amount of slack. What is the largest cube that can pass under the ribbon ?

Pythagorean Triples II

Age 11 to 16

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

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?

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.

Pythagoras

Age 7 to 14

Pythagoras of Samos was a Greek philosopher who lived from about 580 BC to about 500 BC. Find out about the important developments he made in mathematics, astronomy, and the theory of music.

The Spider and the Fly

Age 14 to 16
Challenge Level

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Generating Triples

Age 14 to 16
Challenge Level

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Cutting a Cube

Age 11 to 14
Challenge Level

A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?

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?

The Medieval Octagon

Age 14 to 16
Challenge Level

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Pareq Calc

Age 14 to 16
Challenge Level

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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.

A Chordingly

Age 11 to 14
Challenge Level

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

The Old Goats

Age 11 to 14
Challenge Level

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . .

Corridors

Age 14 to 16
Challenge Level

A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.

Grid Lockout

Age 14 to 16
Challenge Level

What remainders do you get when square numbers are divided by 4?

Two Circles

Age 14 to 16
Challenge Level

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

Are You Kidding

Age 14 to 16
Challenge Level

If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units.... What is the area of the triangle?

Napkin

Age 14 to 16
Challenge Level

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

Medallions

Age 14 to 16
Challenge Level

Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?

Liethagoras' Theorem

Age 7 to 14

Liethagoras, Pythagoras' cousin (!), was jealous of Pythagoras and came up with his own theorem. Read this article to find out why other mathematicians laughed at him.

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?

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. . . .

Slippage

Age 14 to 16
Challenge Level

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .

Where to Land

Age 14 to 16
Challenge Level

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

The Fire-fighter's Car Keys

Age 14 to 16
Challenge Level

A fire-fighter needs to fill a bucket of water from the river and take it to a fire. What is the best point on the river bank for the fire-fighter to fill the bucket ?.

Take a Square

Age 14 to 16
Challenge Level

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

Circle Packing

Age 14 to 16
Challenge Level

Equal circles can be arranged so that each circle touches four or six others. What percentage of the plane is covered by circles in each packing pattern? ...

Squaring the Circle and Circling the Square

Age 14 to 16
Challenge Level

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

Three Cubes

Age 14 to 16
Challenge Level

Can you work out the dimensions of the three cubes?

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.

Three Four Five

Age 14 to 16
Challenge Level

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

Crescents and Triangles

Age 14 to 16
Challenge Level

Can you find a relationship between the area of the crescents and the area of the triangle?

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.

Trice

Age 11 to 14
Challenge Level

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

Isosceles

Age 11 to 14
Challenge Level

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

Floored

Age 14 to 16
Challenge Level

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?