Resources tagged with: Circle properties and circle theorems

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Broad Topics > Angles, Polygons, and Geometrical Proof > Circle properties and circle theorems

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?

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?

Coins on a Plate

Age 11 to 14
Challenge Level

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

Medallions

Age 14 to 16
Challenge Level

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

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.

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?

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

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?

Circumspection

Age 14 to 16
Challenge Level

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

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?

Arrh!

Age 14 to 16
Challenge Level

Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. . . .

Sitting Pretty

Age 14 to 16
Challenge Level

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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?

Cyclic Quad Jigsaw

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?

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?

Not So Little X

Age 11 to 14
Challenge Level

Two circles are enclosed by a rectangle 12 units by x units. The distance between the centres of the two circles is x/3 units. How big is x?

Bicentric Quadrilaterals

Age 14 to 16
Challenge Level

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

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?

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.

Circle Scaling

Age 14 to 16
Challenge Level

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

Circle Theorems

Age 14 to 16
Challenge Level

This set of resources for teachers offers interactive environments which support work on properties of angles in circles at Key Stage 4.

Tricircle

Age 14 to 16
Challenge Level

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and. . . .

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.

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.

Tied Up

Age 14 to 16 Short
Challenge Level

How much of the field can the animals graze?

The Cyclic Quadrilateral

Age 11 to 16

This gives a short summary of the properties and theorems of cyclic quadrilaterals and links to some practical examples to be found elsewhere on the site.

Circles in Quadrilaterals

Age 14 to 16
Challenge Level

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Interacting with the Geometry of the Circle

Age 5 to 16

Jennifer Piggott and Charlie Gilderdale describe a free interactive circular geoboard environment that can lead learners to pose mathematical questions.

Cyclic Quadrilaterals

Age 11 to 16
Challenge Level

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Triangle Incircle Iteration

Age 14 to 16
Challenge Level

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

Triangles in Circles

Age 11 to 14
Challenge Level

Can you find triangles on a 9-point circle? Can you work out their angles?

Subtended Angles

Age 11 to 14
Challenge Level

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

Right Angles

Age 11 to 14
Challenge Level

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Perfect Eclipse

Age 14 to 16
Challenge Level

Use trigonometry to determine whether solar eclipses on earth can be perfect.