STEPS
To prove that the area is 36-6c square units the steps are as follows:
g. a+b+c=12
All sides added together is the perimeter.
c. a+b=12-c
It's the same equation rearranged so you can find out the value of a+b.
f. By Pythagoras theorem, a² + b² = c²
a. Square both sides = a² + 2ab+ b² = 144 - 24c + c²
square both sides of a+b=12-c because of Pythagoras' Theorem, and you do that by:
square a +(a times b + a times b)+ square b = square 12 -(12 times c -12 times c)+ square c
(a²) + (2x a times b = 2ab) + (b²) = (144) - (2x 12 times c = 24c ) + (c²)
d. So 2ab=144-24c
Get rid of a²+b²=c² from the equation, from the previous step so 2ab=144-24c.
h. Dividing by 2: ab=72-12c
Divide it by 2 so you can get the value of ab
e. Area of the triangle = ab/2
The area triangle is half of the area of a rectangle
Length times Width = Area of a rectangle, divided by 2 = Area of a triangle
b. So the area of the triangle = 36-6c
ab/2 = area of the triangle so 72-12c divided by 2 = 36-6c
ADAPTING the method to prove when the perimeter is 30 units the area is 225-15c=
1 a+b+c=30
2 a+b=30-c
3 a² + 2ab+ b² = 900 - 60c + c²
4 So 2ab= 900 - 60c
5 Dividing by 2: ab=350-30c
6 ab/2 is the area of the triangle, therefore the area is 225-15c
EXTENSION: Hypotenuse=c Perimeter=p
The beginning equation is always a+b+c=p
Then you rearrange it and turn it into a+b=p-c
You must then square both sides to make the equation a² + 2ab+ b² = p² - 2pc + c²
Which makes 2ab = p² - 2pc
To get to the area you then divide it by 2 twice, so to simplify it you could just divide it by 4
So a general equation to find out what the area of a right-angled triangle is:
p² - 2pc
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