Determine hypotenuse of right triangle if its area is 6 cm^2 and perimeter is 12 cm?
2 Answers
justaguide | College Teacher | (Level 2) Distinguished Educator
Let the smaller sides of the triangle have a length a and b. The length of the hypotenuse is sqrt( a^2 + b^2)
The area of the triangle is (1/2)*a*b = 6
=> ab = 12
=> a = 12/b
The perimeter is a + b + sqrt (a^2 + b^2) = 12
a + b + sqrt (a^2 + b^2) = 12
replace 12 with ab
=> a + b + sqrt (a^2 + b^2) = ab
=> ab - (a + b) = sqrt (a^2 + b^2)
square the two sides
=> a^2*b^2 + (a +b)^2 - 2ab*(a + b) = a^2 + b^2
=> a^2*b^2 + a^2 + b^2 + 2ab - 2ab*(a + b) = a^2 + b^2
=> a^2*b^2 + 2ab - 2ab*(a + b) = 0
divide all the terms by ab
=> ab + 2 - 2(a + b) = 0
substitute a = 12/b
=> b*12/b + 2 - 2(12/b + b) = 0
=> 14 = 2*(12/b + b)
=> 7 = 12/b + b
=> 7b = 12 + b^2
=> b^2 - 7b + 12 = 0
=> b^2 - 4b - 3b + 12 = 0
=> b(b - 4) - 3(b - 4) = 0
=> (b - 3)(b - 4) = 0
b = 3 or b = 4
a = 4 or a = 3
The hypotenuse is sqrt (a^2 + b^2) = sqrt ( 3^2 + 4^2) = sqrt 25 = 5
The required hypotenuse = 5.
giorgiana1976 | College Teacher | (Level 3) Valedictorian
We'll put the length of the legs of the right triangle as x and y.
A = x*y/2
6 = x*y/2 => x*y = 12 (1)
Since the perimeter is the sum of the lengths of the legs and hypotenuse of right angle triangle, we'll get
P = x + y + sqrt(x^2 + y^2)
Hypothenuse = sqrt(x^2 + y^2) (Pythagorean identity)
12 = x + y + sqrt(x^2 + y^2) (2)
We'll equate (1) and (2):
xy = x + y + sqrt(x^2 + y^2)
We'll subtract both sides x + y:
xy - (x+y) = sqrt(x^2 + y^2)
We'll raise to square both sides:
[xy - (x+y)]^2 = [sqrt(x^2 + y^2)]^2
(xy)^2 + x^2 + 2xy + y^2 - 2xy(x+y) = x^2 + y^2
We'll eliminate x^2 + y^2:
(xy)^2 + 2xy - 2xy(x+y) = 0
We'll factorize by xy:
xy(xy + 2 - 2x - 2y) = 0
We'll put xy = 0
Since xy = 0 => 2 - 2x - 2y = 0
-2(x+y) = -2
x + y = 1
We'll create the quadratic equation whose sum is 1 and product is 0:
x^2 - x = 12
x^2 - x - 12 = 0
x1 = [1+sqrt(1 + 48)]/2
x1 = (1+7)/2
x1 = 4
x2 = -3
Since a length of a side cannot be negative, we'll reject x = -3.
We'll put x = 4
6 = 4*y/2
12 = 4*y
y = 3
The lengths of the legs of the triangle are x = 4 and y = 3.
The hypothenuse is:
H = sqrt(3^2 + 4^2)
H = sqrt25
H = 5 cm
The length of hypotenuse is H = 5 cm.