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There are **59** NRICH Mathematical resources connected to **Similarity and congruence**, you may find related items under Angles, polygons, and geometrical proof.

Problem
Primary curriculum
Secondary curriculum
### The Square Under the Hypotenuse

Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangle in a Trapezium

Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Same Length

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Kite in a Square

Can you make sense of the three methods to work out what fraction of the total area is shaded?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fit for Photocopying

Explore the relationships between different paper sizes.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagoras Proofs

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

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Partly Circles

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Matching Triangles

Can you sort these triangles into three different families and explain how you did it?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Nicely Similar

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### All about Ratios

A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Trapezium Four

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangle in a Triangle

Can you work out the fraction of the original triangle that is covered by the inner triangle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Figure of Eight

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 ?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Squirty

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Hex

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.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Angle Trisection

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Napkin

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sitting Pretty

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Ladders

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 wall. At what height do the ladders cross?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Points in Pairs

Move the point P to see how P' moves. Then use your insights to calculate a missing length.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Von Koch Curve

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Squareflake

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sierpinski Triangle

What is the total area of the triangles remaining in the nth stage of constructing a Sierpinski Triangle? Work out the dimension of this fractal.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pentabuild

Explain how to construct a regular pentagon accurately using a straight edge and compass.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Slippage

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 up the wall which the ladder can reach?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Bus Stop

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 and in the ratio 5 to 4. The buses travel to and fro between the towns. What milestones are at Shipton and Veston?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### All Tied Up

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Flower

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Arrh!

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. What is the value of r/R?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Triangles in a Square

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Shade Crossed

Find the area of the shaded region created by the two overlapping triangles in terms of a and b?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pentakite

Given a regular pentagon, can you find the distance between two non-adjacent vertices?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Chord

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Overlap

A red square and a blue square overlap. Is the area of the overlap always the same?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangular Tantaliser

Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Parallel Universe

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Matter of Scale

Can you prove Pythagoras' Theorem using enlargements and scale factors?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pent

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.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Folding Squares

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### How Big?

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### From All Corners

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Sameness Surely

Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Look Before You Leap

Can you spot a cunning way to work out the missing length?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Double Angle Triples

Try out this geometry problem involving trigonometry and number theory

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Strange Rectangle

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fitting In

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 equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Golden Triangle

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Is a Square a Rectangle?

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

Age 7 to 11

Challenge Level