Triangles Investigation

Triangles Investigation

“Two students are discussing how to find the biggest value of the

area: perimeter ratio for triangles. One of them suggests that this

can be done with measurements of 40, 60 and 80 but forgets to say what

units were used and whether they were angles or sides. Which triangle

gives the biggest area: perimeter ratio? Investigate further.”

To investigate this problem I am going to find out all the different

combinations of triangles with the measurements of 40, 60 and 80. Then

work out their area: perimeter ratio. From this I hope to make a valid

conclusion.

1) If 40, 60 and 80 are all sides;

[IMAGE]

[IMAGE]

Therefore the perimeter = 40+60+80=180

And the area, which can be worked out using the formula:

1/2absinC = 0.5x40x80xsin47 = 1170.16…**

So the area: perimeter ratio = 1170.16… / 180 = 6.5 (1dp)

** The … means there are more numbers but I haven’t rounded down to

make sure my final answer is more accurate.

2) If 40, 60 and 80 are all angles;

[IMAGE]

I cannot work out the ratio for this triangle using numbers. This is

because I don’t have enough information to work out sides. I will come

back to this triangle later.

3) If there are 2 angles and 1 side;

This is just a specific case of 2 and I will deal with this later. It

is a specific case because if I have the angles 40o and 60o the other

angle will have to be 80o for all of them to equal 180o.

4) If there are 2 sides and 1 angle;

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To work out this triangle I need to use the cosine formula: c2=a2+b2-2abcosC

The three combinations of this triangle are:

1-a=60, b=80, C*=40

2-a=40, b=80, C=60

3-a=40, b=60, C=80

a

The angle is the included angle;

[IMAGE]

*C being the angle

1- Perimeter = 60+80+51=191

Area = 1542.