Resources tagged with: Pythagoras' theorem

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Broad Topics > Pythagoras and Trigonometry > Pythagoras' theorem

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.

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.

Get Cross

Age 14 to 16
Challenge Level

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

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?

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?

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.

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!

Pythagorean Triples II

Age 11 to 16

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

Some(?) of the Parts

Age 14 to 16
Challenge Level

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

Compare Areas

Age 14 to 16
Challenge Level

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?

Square Pegs

Age 11 to 14
Challenge Level

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

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?

Holly

Age 14 to 16
Challenge Level

The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.

Hex

Age 11 to 14
Challenge Level

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.

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.

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?

Grid Lockout

Age 14 to 16
Challenge Level

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

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?

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.

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?

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. . . .

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?

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.

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.

Semi-detached

Age 14 to 16
Challenge Level

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

Circumnavigation

Age 14 to 16
Challenge Level

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

Partly Circles

Age 14 to 16
Challenge Level

What is the same and what is different about these circle questions? What connections can you make?

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? ...

Medallions

Age 14 to 16
Challenge Level

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

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?

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?

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.

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 ?.

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. . . .

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?

Squareo'scope Determines the Kind of Triangle

Age 11 to 14

A description of some experiments in which you can make discoveries about triangles.

Pythagoras Proofs

Age 14 to 16
Challenge Level

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

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?

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?

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?

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?

Circle Scaling

Age 14 to 16
Challenge Level

Describe how to construct three circles which have areas in the ratio 1:2:3.

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 ?

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?

Matter of Scale

Age 14 to 16
Challenge Level

Prove Pythagoras' Theorem using enlargements and scale factors.

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.

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.

Ball Packing

Age 14 to 16
Challenge Level

If a ball is rolled into the corner of a room how far is its centre from the corner?