Resources tagged with: Pythagoras' theorem

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Broad Topics > Pythagoras and Trigonometry > Pythagoras' theorem

Orthogonal Circle

Age 16 to 18
Challenge Level

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.

Baby Circle

Age 16 to 18
Challenge Level

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

Incircles Explained

Age 16 to 18

This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.

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?

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?

Retracircles

Age 16 to 18
Challenge Level

Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.

Logosquares

Age 16 to 18
Challenge Level

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

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 ?

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?

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

Belt

Age 16 to 18
Challenge Level

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.

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. . . .

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?

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.

Strange Rectangle

Age 16 to 18
Challenge Level

ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.

Grid Lockout

Age 14 to 16
Challenge Level

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

Circle Scaling

Age 14 to 16
Challenge Level

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

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?

The Dodecahedron

Age 16 to 18
Challenge Level

What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?

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.

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?

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.

Incircles

Age 16 to 18
Challenge Level

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

Medallions

Age 14 to 16
Challenge Level

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

Square World

Age 16 to 18
Challenge Level

P is a point inside a square ABCD such that PA= 1, PB = 2 and PC = 3. How big is angle APB ?

Xtra

Age 14 to 18
Challenge Level

Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.

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.

Inscribed in a Circle

Age 14 to 16
Challenge Level

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

Chord

Age 16 to 18
Challenge Level

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

At a Glance

Age 14 to 16
Challenge Level

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

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. . . .

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?

Golden Construction

Age 16 to 18
Challenge Level

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

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.

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? ...

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?

Far Horizon

Age 14 to 16
Challenge Level

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

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?

Babylon Numbers

Age 11 to 18
Challenge Level

Can you make a hypothesis to explain these ancient numbers?

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.

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. . . .

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.

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.

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 ?.

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.

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?

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 .

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.