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.
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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?
The diagonals of a square meet at O. The bisector of angle OAB meets BO and BC at N and P respectively. The length of NO is 24. How long is PC?
Using a ruler, pencil and compasses only, it is possible to
construct a square inside any triangle so that all four vertices
touch the sides of the triangle.
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.
The points P, Q, R and S are the midpoints of the edges of a convex
quadrilateral. What do you notice about the quadrilateral PQRS as
the convex quadrilateral changes?
On a nine-point pegboard a band is stretched over 4 pegs in a
"figure of 8" arrangement. How many different "figure of 8"
arrangements can be made ?
Photocopiers can reduce from A3 to A4 without distorting the image.
Explore the relationships between different paper sizes that make
Make a poster using equilateral triangles with sides 27, 9, 3 and 1 units assembled as stage 3 of the Von Koch fractal. Investigate areas & lengths when you repeat a process infinitely often.
A finite area inside and infinite skin! You can paint the interior
of this fractal with a small tin of paint but you could never get
enough paint to paint the edge.
Straight lines are drawn from each corner of a square to the mid
points of the opposite sides. Express the area of the octagon that
is formed at the centre as a fraction of the area of the square.
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. . . .
Two ladders are propped up against facing walls. The end of the
first ladder is 10 metres above the foot of the first wall. The end
of the second ladder is 5 metres above the foot of the second. . . .
Two intersecting circles have a common chord AB. The point C moves
on the circumference of the circle C1. The straight lines CA and CB
meet the circle C2 at E and F respectively. As the point C. . . .
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?
What is the total area of the triangles remaining in the nth stage
of constructing a Sierpinski Triangle? Work out the dimension of
Three triangles ABC, CBD and ABD (where D is a point on AC) are all
isosceles. Find all the angles. Prove that the ratio of AB to BC is
equal to the golden ratio.
Explain how to construct a regular pentagon accurately using a
straight edge and compass.
Find the area of the shaded region created by the two overlapping
triangles in terms of a and b?
The diagram shows a regular pentagon with sides of unit length.
Find all the angles in the diagram. Prove that the quadrilateral
shown in red is a rhombus.
Six circles around a central circle make a flower. Watch the flower
as you change the radii in this circle packing. Prove that with the
given ratios of the radii the petals touch and fit perfectly.
Given that ABCD is a square, M is the mid point of AD and CP is
perpendicular to MB with P on MB, prove DP = DC.
What fractions can you divide the diagonal of a square into by simple folding?
Plex lets you specify a mapping between points and their images.
Then you can draw and see the transformed image.
The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?
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?
Two tangents are drawn to the other circle from the centres of a
pair of circles. What can you say about the chords cut off by these
tangents. Be patient - this problem may be slow to load.
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. . . .
Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant. . . .