Resources tagged with: Pythagoras' theorem

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Broad Topics > Pythagoras and Trigonometry > Pythagoras' theorem

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.

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!

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 ...?

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?

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?

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?

Medallions

Age 14 to 16
Challenge Level

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

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.

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?

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?

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 II

Age 11 to 16

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

Pythagoras Proofs

Age 14 to 16
Challenge Level

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

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?

Squ-areas

Age 14 to 16
Challenge Level

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .

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.

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.

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 ?

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.

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? ...

Circle Scaling

Age 14 to 16
Challenge Level

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

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?

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?

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. . . .

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.

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?

Six Discs

Age 14 to 16
Challenge Level

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

Grid Lockout

Age 14 to 16
Challenge Level

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

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?

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?

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 .

Rectangular Pyramids

Age 14 to 18
Challenge Level

Is the sum of the squares of two opposite sloping edges of a rectangular based pyramid equal to the sum of the squares of the other two sloping edges?

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.

30-60-90 Polypuzzle

Age 16 to 18
Challenge Level

Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.

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.

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.

Matter of Scale

Age 14 to 16
Challenge Level

Prove Pythagoras' Theorem using enlargements and scale factors.

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.

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.

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.

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.

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?

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?

Reach for Polydron

Age 16 to 18
Challenge Level

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

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 ?

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.

Babylon Numbers

Age 11 to 18
Challenge Level

Can you make a hypothesis to explain these ancient numbers?

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?