equable right angled triangles

not using pythagoras

working out the radius of the incircle

three little triangles sum to the large one

an alternative method, using pythagoras, to find the radius of the incircle

proving that there are just two right angled equable triangles with integer lengths

two of the possible values duplicate the other two

equable isosceles triangles

the area value = perimeter value
for isosceles triangles (including an equilateral triangle, in Q2) with integer heights

this work involves: surds,
rationalising (simple) denominators
and pythagoras

three triangles sum to the large (isosceles) triangle

and 2a + 2c = ah since P = A

fractions of rectangles

this is practice in areas of shapes
with the question reversed - given the area, what could the shape look like?

students will hopefully seek suitable base and height dimensions
(possibly forgetting that in a triangle the area involves halving)

some questions have two solutions
this can create opportunities to discuss why e.g. triangles with the same base and between two parallels have the same area

you could include blank grids on the back of each sheet

these could involve finding areas by dissecting a rectangle

the bottom left two can be justified visually

or you could involve surds

quadrilaterals with 3 acute angles

the initial question is from a KS2 test paper (2016)
the tasks are good practice in using areas of triangles
the powerpoint is here

triangle areas, various ways

rather big numbers (use a calculator)
the intention, advocated by Ed Southall (solve my maths), is for students to decide which dimensions they need in order to work out the area of a triangle

then they can be asked to calculate it another way
(and possibly a third way)

the areas are:
(1) 150
(2) 5,070
(3) 302,580
(4) 1,227,930

triangles cut into triangles

the main purpose of this work is to provide a focus on triangle areas
particularly with same bases and equal heights (Euclid I. 37)
but there are various optional spin-offs:

• perpendicular lines
• surd lengths
• length and area ratios of similar shapes

 

questions for the triangles above

the team at Puntmat (@puntmat) have found 8 ways (so far) to cut up a 3 by 6 right triangle into three equal areas (i.e. non-congruent thirds):

their post on fractions of geometrical shapes contains many good exercises

many thanks to them











dividing unequally:

rhombus area

using areas of triangles to work out areas of rhombuses
possibly generalising

did the Babylonians use similar triangles?

IM55357 is a tablet showing right angled triangles

it is sometimes used as evidence that the Babylonians (around 1900 to 1600BCE) were able to use similar triangle techniques

EM Bruins points out that there is not widespread evidence of this being the case

he shows that a relationship can be derived just by using areas of triangles and a trapezium

how?

area is 8 squares