Explanation:
right-angled triangle.
that means Pythagoras applies :
c² = a² + b²
c being the Hypotenuse (the side opposite of the 90° angle and the longest of the 3 sides).
(x+4)² = (x+1)² + (x+2)²
x² + 8x + 16 = x² + 2x + 1 + x² + 4x + 4
8x + 16 = x² + 6x + 5
x² - 2x = 11
remember,
(x-a)² = x² -2ax + a²
compare to what we have
-2a = -2
a = 1
so, the full square on the left side is
(x-1)² = x² - 2x + 1
to complete the square in our equation we need to add 25 on both sides
x² - 2x + 1 = 11 + 1 = 12
(x - 1)² = 12
x - 1 = sqrt(12)
x = sqrt(12) + 1
and so
i)
when we have all 3 sides a, b, c, the area of the triangle is according to Heron's formula
S = (a + b + c)/2
Area = sqrt(S(S-a)(S-b)(S-c))
in our case that is
S = ((x+1) + (x+2) + (x+4))/2 =
= ((sqrt(12)+1+1) + (sqrt(12)+1+2) + (sqrt(12)+1+4))
/2 = (3×sqrt(12) + 10)/2 =
= 10.19615242...
Area = sqrt(S(S- sqrt(12)-2)(S-sqrt(12)-3)(S-sqrt(12)-5)) =
= sqrt(10.19615242... ×
4.732050808... ×
3.732050808... ×
1.732050808...) =
= sqrt(311.8845727...) =
= 17.66025404... cm²
ii)
the perimeter is
(sqrt(12)+1+1) + (sqrt(12)+1+2) + (sqrt(12)+1+4) =
3sqrt(12) + 10 = 20.39230485... cm