Skip over navigation
Cambridge University Faculty of Mathematics NRich logo
menu search
  • Teachers expand_more
    • Early years
    • Primary
    • Secondary
    • Post-16
    • Events
    • Professional development
  • Students expand_more
    • Primary
    • Secondary
    • Post-16
  • Parents expand_more
    • Early Years
    • Primary
    • Secondary
    • Post-16
  • Problem-Solving Schools
  • About NRICH expand_more
    • About us
    • Impact stories
    • Support us
    • Our funders
    • Contact us
  • search

Or search by topic

Number and algebra

  • The Number System and Place Value
  • Calculations and Numerical Methods
  • Fractions, Decimals, Percentages, Ratio and Proportion
  • Properties of Numbers
  • Patterns, Sequences and Structure
  • Algebraic expressions, equations and formulae
  • Coordinates, Functions and Graphs

Geometry and measure

  • Angles, Polygons, and Geometrical Proof
  • 3D Geometry, Shape and Space
  • Measuring and calculating with units
  • Transformations and constructions
  • Pythagoras and Trigonometry
  • Vectors and Matrices

Probability and statistics

  • Handling, Processing and Representing Data
  • Probability

Working mathematically

  • Thinking mathematically
  • Developing positive attitudes
  • Cross-curricular contexts

Advanced mathematics

  • Decision Mathematics and Combinatorics
  • Advanced Probability and Statistics
  • Mechanics
  • Calculus

For younger learners

  • Early Years Foundation Stage

Isosceles

Age 11 to 14
Challenge Level Yellow starYellow starYellow star
Secondary curriculum
  • Problem
  • Student Solutions

From the diagram sent in by Chin Siang, Tao Nan School, Singapore, you can see that triangles AEG and BDI are equal in area because each is made up of two right angles triangles with sides 3, 4 and 5 units with the line IC as an axis of symmetry:


Suzanne and Bethany (the Mount School, York) sent us this solution:

Any isosceles triangle with base b , height h and area bh /2 can be split into two right angled triangles along the axis of symmetry and these right angled triangles can be re-assembled into a second isosceles triangle with base 2 h , height b /2 and area bh /2. The only case in which the pair of isosceles triangles are congruent is where b = 2 h so there are infinitely many pairs of non congruent isosceles triangles with the same area. In general the sides are not of integer length but there are also infinitely many such pairs with sides of integer length formed from right angled in which the lengths of the sides are Pythagorean triples. For example the two isosceles triangles with sides 13,13, and 10 units and with sides 13,13, and 24 units both have area 60 square units.

The Pythagorean triple 7, 24, 25 gives triangles with sides 25, 25, 14 and 25, 25, 48 which both have areas of 168 square units, and we can go on doing this with Pythagorean triples of which there are lots. As for isosceles triangles with an area of 12 square units, well it only needs the base and height to have a product of 24, so, if we insist on whole numbered sides, then there are only two - the ones given. If sides do not have to be whole numbers then there are an infinite number of solutions, though some would be difficult to tell apart.


You may also like

Square Pegs

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

The Old Goats

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?

Trice

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

  • Tech help
  • Accessibility Statement
  • Sign up to our newsletter
  • Twitter X logo

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

University of Cambridge logo NRICH logo