Find all sides of a right triangle whose perimeter is equal to 60 cm and its area is equal to 150 square cm.
Answers
x + y + H = 60 : perimeter , x, y and H be the two legs and the hypotenuse of the right triangle
(1/2)xy = 150 : area
x2 + y2 = H2: Pythagora's theorem.
3 equations with 3 unknowns.
(x + y)2 - 2xy = H2 : completing the square in the third equation.
x + y = 60 - H : express x + y using the first equation and use the second equation to find xy = 300 and substitute in equation 5.
(60 - H)2 - 600 = H2 : one equation with one unknown.
Solve for H to find H = 25 cm. Substitute and solve for x and y to find x = 15 cm and y = 20 cm.
Answer:
Step-by-step explanation:
Let us assume that b,hand H are sides of the right triangle.
Then,
base = b
height = h
hypotenuse = H
b² + h² = H²
⇒(b+h)² - 2bh = H²
Area = 150
⇒(1/2)bh = 150
⇒bh = 300
Perimeter = 60
⇒ b + h + H = 60
⇒b + h = 60-H
Thus (b+h)² - 2bh = H²
⇒(60-H)² -2×300 = H²
⇒ 3600 + H² - 2×60×H - 2×300 = H²
⇒3600 - 120H -600 = 0
⇒ -120H + 3000 = 0
⇒120H = 3000
⇒ H = 3000/120 = 25cm
b+h = 60-25 = 35
⇒300/h + h = 35
⇒h² + 300 = 35h
⇒h² -35h + 300 = 0
⇒ h² -15h -20h +300 = 0
⇒(h-15)(h-20) = 0
⇒h = 15 or 20
⇒b = 20 or 15
Hope it will help you