Question

Find all sides of a right triangle whose perimeter is equal to 60 cm and its area is equal to 150 square cm. ​

Answer 1

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 2

Solution:-

Given:-

Perimeter of Right Triangle = 60cm.

Area of Right Triangle = 150 cm²

To Find:-

All the sides of the Triangle.

Find:-

Let the three sides of the triangle be H , b and h.

Where,

H = Hypotenuse.

h = Height.

b = base.

Since, It's a Right Triangle.

=) By Pythagoras Theorem,

b² + h² = H²

⇒(b+h)² - 2bh = H² [ (a + b)² -2ab = a² + b² ]______________(1)

Now,

Area of Triangle = 150 cm².

⇒(1/2)bh = 150

⇒bh = 300

⇒ b = 300/h___________(2)

Perimeter of Triangle = 60 cm

⇒ b + h + H  = 60

⇒b + h = 60-H__________(3)

Substituting [ b + h = 60 - H ] in eq (1). we get,

⇒(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

⇒ H = 25cm

Substituting [ H = 25cm ] in eq (3).

⇒ b + h = 60 - 25

⇒ b + h = 35

⇒300/h + h = 35 [ from eq(2) ]

⇒ ( 300 + h²)/h = 35

⇒h² + 300 = 35h

⇒h² -35h + 300 = 0

⇒ h² -15h -20h +300 = 0

⇒(h-15)(h-20) = 0

⇒h = 15 or 20

So,

If h = 15cm then b = 20 cm

OR

If h = 20cm then b = 15 cm

Hence,

Three sides are 15cm , 20cm and 25cm.