Question

the perimeter of a right angled triangle is40 cm if It's hypotenuse is 17 cm find area
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Answer 1

Given,

⟹ a + b + c = 40cm

⟹ b = 17cm

Solution:

⇒ a + 17 + c = 40

⇒ a + c = 23 --(i)

⇒ b² = a² + c²

(Pythagoras theorem)

⇒ (17)² = a² + c²

⇒ a² + c² = 289 --(ii)

⇒ (a + c)² = a² + c² + 2ac

⇒ (23)² = 289 + 2ac

⇒ ac = 120 --(iii)

∴ Area = 1/2 × base × height

= 1/2 × a × c

= 1/2 × 120

= 60 cm²

Answer 2

GIVEN :-

Perimeter of right angled ∆ = 40 cm.

Hypotenuse of right angled ∆ = 17 cm.

TO FIND :-

The area of the right angled ∆.

SOLUTION :-

Let, the Perpendicular of ∆ be "x" and the base of the triangle be "y".

Now , as we know that,

⇒ Perimeter of ∆ = sum of all sides.

⇒ Hypotenuse + x + y = 40 cm.

⇒ 17 cm + x + y = 40 cm

⇒ x + y = 40 cm - 17 cm.

⇒ x + y = 23

⇒ x = 23 - y. ....(1)

Now by using the Pythagoras theorem we have,

⇒ (Hypotenuse)² = (x)² + (y)²

⇒ (17)² = (23 - y)² + (y)²

⇒ 289 = (23)² + (y)² - 2 × 23 × y + (y)²

⇒ 289 = 529 + y² - 46y + y²

⇒ 289 = 529 + y² + y² - 46y

⇒ 289 - 529 = y² + y² - 46y

⇒ -240 = y² + y² - 46y

⇒ -240 = 2y² - 46y

⇒ 2y² - 46y + 240 = 0

⇒ 2(y² - 23y + 120) = 0

⇒ y² - 23y + 120 = 0/2

⇒ y² - 23y + 120 = 0

By splitting the middle term process,

⇒ y² - 15y - 8y + 120 = 0

⇒ y(y - 15) - 8(y - 15)

⇒ (y - 15) (y - 8) = 0

⇒ y - 15 = 0 , y - 8 = 0

⇒ y = 15 , y = 8

Now substitute the value of y in equation 1.

⇒ x = 23 - y [ ∵ y = 15 , y = 8 ]

⇒ x = 23 - 15. [ ∵ y = 15 ]

⇒ x = 8

Similarly,

⇒ x = 23 - y

⇒ x = 23 - 8. [ ∵ y = 8 ]

⇒ x = 15

Now as we know that,

⇒ Area of triangle = 1/2 × base × Height

⇒ Area of triangle = 1/2 × 15 × 8

⇒ Area of triangle = 15 × 4

⇒ Area of triangle = 60 cm².

On taking the other values,

⇒ Area of triangle = 1/2 × base × Height

⇒ Area of triangle = 1/2 × 8 × 15

⇒ Area of triangle = 4 × 15

⇒ Area of triangle = 60 cm².

⇒ Area of triangle = 60 cm.Hence the area of the triangle is 60 cm².